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hra^uL  ,/^    e-^*-c/2v^^.^<^^^^-  ^ 


UNIPLANAR   ALGEBRA 


PART  I   OF  A  PROPEDEUTIC  TO  THE  HIGHER 
MATHEMATICAL  ANALYSIS 


IRVING   STRINGHAM,  Ph.  D. 

Professor  of  Mathematics  in  the  University  of  California 


San  Francisco 
THE    BERKELEY    PRESS 

I  So-! 


CCPVRIGHT,    1893 
BY 

IRVING   STRINGHAM 


Typography  and  Presswork  by  C.  A,  MURDOCK  ^-  Co  ,  San  Francisco 


CMS 


PREFACE. 


From  the  beginning,  with  rare  exceptions,  -^  a  singular 
logical  incompleteness  has  characterized  our  text-books  in 
elementary  algebra.  By  tradition  algebra  early  became  a 
mere  technical  device  for  turning  out  practical  results,  by 
careless  reasoning  inaccuracies  crept  into  the  explanation  of 
its  principles  and,  through  compilers,  are  still  perpetuated 
as  current  literature.  Thus,  instead  of  becoming  a  classic, 
like  the  geometry  handed  down  to  us  from  the  Greeks,  in 
the  form  of  Euclid's  Elemeiits,  algebra  has  become  a  collec- 
tion of  processes  practically  exemplified  and  of  principles 
inadequately  explained. 

The  labors  of  the  mathematicians  of  the  nineteenth 
century  —  Argand,  Gauss,  Cauchy,  Grassmann,  Peirce, 
Cayley,  Sylvester,  Kronecker,  Weierstrass,  G.  Cantor, 
Dedekind  and  others*-!^  —  have  rendered  unjustifiable  the 
longer  continuance  of  this  unsatisfactory  state  of  algebraic 
science.  We  now  know  what  an  algebra  is,  and  the  prob- 
lem of  its  systematic  unfolding  into  organic  form  is  a  definite 
and  achievable  one.  The  short  treatise  here  presented,  as 
the  first  part  of  a  Propaedeutic  to  the  Higher  Analysis, 
endeavors  to  place  concisely  in  connected  sequence  the 
argument  required  for  its  solution. 


*  Notably,  in  English,  Chrystal's  Algebra,  2  vols.,  Edinburgh,  1886,  18S9.  Oti 
the  continent  of  Europe  the  deficiency  has  been  compensated  mainly  in  works  on 
the  Higher  Analysis. 

**The  literature  through  which  algebra  has  been  rehabilitated  during  the 
present  century  is  extensive.  See  Stolz:  AUgemeuie  A^itlnnetik,  Leipzig,  1885, 
for  many  valuable  references. 


IV  PREF-ACE. 

The  first  three  chapters  were  made  pubHc,  substantially 
in  their  present  form,  in  a  course  of  University  Extension 
Lectures  in  San  Francisco  during  the  winter  of  1891-92,  a 
synopsis  of  which  was  issued  from  the  University  Press  in 
October,  1S91.  At  the  close  of  these  lectures  the  manu- 
script of  the  complete  work  was  prepared  for  the  press ;  but 
unavoidable  obstacles  prevented  its  immediate  publication 
and  a  consequent  delay  of  somewhat  more  than  a  year  has 
intervened.  This  delay,  however,  has  made  possible  a 
revision  of  the  original  sketch  and  some  additions  to  its 
subject-matter. 

The  logical  grounding  of  algebra  may  be  attained  by 
either  of  two  methods,  the  one  essentially  arithmetical,  the 
other  geometrical,  I  have  chosen  the  geometrical  form  of 
presentation  and  development,  partly  because  of  its  simpler 
elegance,  partly  because  this  way  lies  the  shortest  path  for 
the  student  who  knows  only  the  elements  of  geometry  and 
algebra  as  taught  in  our  schools  and  requires  mathematical 
study  only  for  its  disciplinary  value.  The  choice  of  method, 
therefore,  is  not  to  be  interpreted  to  mean  that  the  writer 
underestimates  the  value  and  the  importance  to  the  special 
mathematical  student  of  the  Number-System.*  This 
system,  however,  has  no  appropriate  place  in  the  plan  here 
presented. 

The  point  of  departure  is  EucHd's  doctrine  of  proportion, 
and  the  point  of  view  is  the  one  that  Euclid  himself,  could 
he  have  anticipated  the  modern  results  of  mathematical 
science,  would  naturally  have  taken.  It  is  interesting  to 
note  that  of  logical  necessity  the  development  falls  mainly 
into  the  historical  order.  For  convenience  of  reference  the 
fundamental  propositions  of  proportion  are  enunciated  and 
proved  in  an  Introduction,   in  which  I  have  followed  the 


*  Fine :  The  Nmnber'System  of  Algebra,  Boston,  1S91. 


PREFACE. 


method  recommended  by  the  Association  for  the  Impro\-e- 
ment  of  Geometrical  Teaching,  and  pubhshed  in  its 
Syllabus  of  Plane  Geometry.  Except  a  few  additions  and 
omissions,  the  enunciations  and  numbering  in  Sections  B 
and  C  of  this  Introduction  are  those  of  Hall  and  Stevens' 
admirable  Text- Book  of  Euclid's  Elejuents,  Book  V;  and 
in  Section  D  those  of  the  Syllabus  of  Plane  Geometry, 
Book  IV,  Section  2.  The  proofs  vary  in  unessential  par- 
ticulars from  those  of  the  two  texts  named. 

The  subject-matter  and  treatment  are  such  as  to  con- 
stitute, for  the  student  already  familiar  with  the  elements  of 
algebra  and  trigonometry,  a  rapid  review  of  the  underlying 
principles  of  those  subjects,  including  in  its  most  general 
aspects  the  algebra  of  complex  quantities.  All  the  funda- 
mental formulae  of  the  circular  and  hyperbolic  functions  are 
concisely  given.  The  chapter  on  Cyclometry  furnishes, 
presumptively,  a  useful  generalization  of  the  circular  and 
hyperbolic  functions. 

The  generalized  definition  of  a  logarithm  (Art.  68)  and 
the  classification  of  logarithmic  systems,*  first  made  pubHc, 
outside  of  the  mathematical  lecture-room,  in  a  paper  read 
before  the  New  York  Mathematical  Society  in  October, 
1 89 1,  and  subsequently  published  in  the  Americaji  fournal 
of  Mathematics,  are  here  reproduced  in  the  revised  form 
suggested  by  Professor  Haskell.  -^  A  chapter  on  Graphical 
Transformations,  giving  the  orthomorphosis  of  the  ex- 
ponential and  cyclic  functions,  appropriately  concludes  this 
part  of  the  subject. 

Many  incidental  problems  are  suggested  in  the  form 
of  Agenda,  useful  to  the  student  for  exemplification  and 
practice.       But    on    the    other    hand,     many    elementary 


*  American  Journal  '}f  Mathematics,  \o\.  XIV,  pp.   1S7-194,  and  Bulletin  of 
the  New  York  Mathematical  Society  \o\.  II,  pp.  164-170. 


PREFACE. 


algebraic  topics  are  not  discussed,  because  they  are  not 
useful  to  the  main  object  t)f  the  work,  and  it  was  especially 
desirable  that  its  purpose  should  not  be  hindered  by  the 
making  of  a  large  book. 

A  few  innovations  in  notation  and  nomenclature  have 
been  unavoidably  introduced.  The  temptation  to  replace 
the  terms  complex  qica7itity,  imaginary  quantity  and  real 
q2iantity  by  some  such  terms  2iS  go7iio7i,  orthogon  and  agon 
has  been  successfully  resisted. 

Partly  in  order  to  aid  the  student  in  obtaining  a  com- 
parative view  of  the  subject,  partly  in  order  to  indicate  in 
some  detail  the  sources  of  information  and  give  due  credit 
to  other  writers,  numerous  foot-note  references  have  been 
introduced. 

I  take  great  pleasure  in  acknowledging  my  obligations 
to  Professor  Haskell  for  valuable  criticism  and  suggestion. 


IRVING  STRINGHAM. 


University  of  California, 
Berkeley,  July  i,  1893. 


CONTENTS 


INTRODUCTION. 
Euclid's  Doctrine  of  Proportion. 

ARTICLE.  PAGE. 

A.  Notation 3 

B.  Definitions  and  axioms 3 

C.  Paraphrase  ft'om  tiie  fifth  book  of  Euclid 7 

D.  Seven  fundamental  theorems  in  proportion 13 

E.  Agenda:  Supplementary  propositions 20 


CHAPTER   I. 

LAWS  OF  ALGEBRAIC  OPERATION. 

I.     Quantity. 

1.  Quantities  in  general 21 

2.  Nature  of  real  quantities 22 

II.     Definitions  of  Algebraic  Operations. 

3.  Algebraic  addition 23 

4.  Zero 23 

5.  Algebraic  multiplication 23 

6.  The  reciprocal 24 

7.  Idemfactor  :   Real  unit 25 

8.  Quotient 25 

9.  Agenda  :  Problems  in  construction 26 

10.     Infinity 26 

ir.     Indeterminate  algebraic  forms 27 


Vlll  CONTENTS. 

III.     Law  of  Signs  for  Real  Quantities. 

ARTICLE.  PAGE. 

12.  In  addition  and  subtraction 28 

13.  In  multiplication  and  division 29 

14.  In  combination  with  each  other  :    +  ,    —  with    X  >    /  •    •  30 

IV.  Associative  Law  for  Real  Quantities. 

15.  In  addition  and  subtraction 31 

16.  In  multiplication  and  division 32 

V.  Commutative  Law  for  Real  Quantities. 

17.  In  addition  and  subtraction 34 

18.  In  multiplication  and  division 35 

19.  Agenda  :  Theorems  in  proportion 36 

VI.  Distributive  Law  for  Real  Quantities. 

20.  With  the  sign  of  multiplication 36 

21.  With  the  sign  of  reciprocation 37 

22.  Agenda :   Theorems  in  proportion,  arithmetical  multipli- 

cation and  division 38 

VII.  Logarithms  and  Exponentials. 

23.  Definitions:   Napierian  definition  of  a  logarithm  .    ...  39 

24.  Relations  between  base  and  modulus 41 

25.  Law  of  involution 42 

26.  Law  of  metathesis 43 

27.  The  law  of  indices 43 

28.  The  addition  theorem . 44 

29.  Infinite  values  of  a  logarithm 45 

30.  Indeterminate  exponential  forms 45 


CONTENTS.  iX 

VIII.     Synopsis  of  Laws  of  Algebraic  Operation. 

ARTICLE.  PAGE. 

31.     Law  of  Signs 46 

32-36.     Laws  of  association,   commutation,  distribution,  ex- 
ponentiation and  logaritiimication 47 

37.  Properties  of  o,  i  and  00 ^  48 

38.  Agenda  :   Involution  and  logarithmic  operation  in  arith- 

metic    48 


CHAPTER    IL 

GONIOMETRIC  AND   HYPERBOLIC   RATIOS. 

IX.     Goniometric  Ratios. 


Definition  of  arc-ratio 49 

Definitions  of  goniometric  ratios 50 

Agenda:  Properties  of  goniometric  ratios 51 

Line  equivalents  of  goniometric  ratios 51 

Proofthat  limit  (sin  ^)  /  ^=  I,  when  ^  J=o 52 

Area  of  a  circular  sector 53 

Agenda  :  The  addition  theorem  for  goniometric  ratios  .  54 


X.     Hyperbolic  Ratios. 

Definitions  of  hyperbolic  ratios 55 

Agenda  :   Properties  of  hyperbolic  ratios 55 

Geometrical  construction  for  hyperbolic  ratios 55 

Agenda  :   Properties  of  the  equilateral  hyperbola  ....  58 

The  Gudermannian 58 

Agenda  :   Gudermannian  formulae 58 

Proofthatlimit  (sinh?/") /^^  =  i.  when?^  =  o 59 

Area  of  a  hyperbolic  sector 60 

Agenda :  The  addition  theorem  for  hyperbolic  ratios  .    .  62 

Approximate  value  of  natural  base 63 

Agenda:   Logarithmic  forms  of  inverse  hyperbolic  ratios  65 


X  CONTENTS. 

CHAPTER    III. 

THE    ALGEBRA   OF  COMPLEX   QUANTITIES. 

XI.     Geometric  Addition  and  Multiplication. 

ARTICLE.  PAGE. 

57.  Classification  of  magnitudes  :   Definitions 66 

58.  Geometric  addition 68 

59.  Commutative  and  associative  laws  for  geometric  addition  69 

60.  Geometric  multiplication 70 

61.  Conjugate  and  reciprocal 71 

62.  Agenda:   Properties  of cis^ 72 

63.  The  imaginary  unit 72 

64.  Commutative  and  associative  laws  for  geometric  multipli- 

cation    73 

65.  The  distributive  law 74 

66.  Argand's  diagram 75 

67.  Problems  in  complex  quantities 76 

XII.     Exponentials  and  Logarithms. 

68.  Definitions  of  modulus,  base,  exponential,  logarithm  .    .  78 

69.  Exponential  of  o,  land  logarithm  of  I,  i5* 81 

70.  Classification  of  logarithmic  systems 81 

71.  Special  constructions 83 

72.  Correspondence    of   initial    values    of  exponential  and 

logarithm 84 

73.  The  exponential  formula 84 

74.  Demoivre's  theorem 86 

75.  Relations  between  base  and  modulus 86 

76.  The  law  of  involution 88 

77.  The  law  of  metathesis 89 

78.  The  law  of  indices 89 

79.  The  addition  theorem  for  logarithms 90 

80.  The  logarithmic  spiral 90 

81.  Periodicity  of  exponentials 91 

82.  Many-valuedness  of  logarithms 92 

83.  Direct  and  inverse  processes 92 

84.  Agenda:  Reduction  of  exponential  and  logarithmic  forms  94 


CONTENTS.  XI 

XIII.     What  Constitutes  an  Algebra? 

ARTICLE.  PAGE. 

85.  The  cycle  of  operations  complete 95 

86.  Definition  of  an  algebra 96 

XIV.     Numerical  Measures. 

87.  Scale  of  equal  parts 97 

88.  For  real  magnitudes 98 

89.  For  complex  magnitudes 100 


CHAPTER    IV. 

CYCLOMETRY. 

XV.     Cyclic  Functions. 

90.  Definitions  of  modocyclic  functions loi 

91.  Formulae  of  cyclic  functions 102 

92.  Agenda  :   Examples  in  multiples  and  submultiples  of  the 

argument 106 

93.  Periodicity    of    modocyclic,    hyperbolic    and     circular 

functions 107 

94.  Agenda:  Functions  of  submultiples  of  the  periods  .  .    .  107 

95.  The  inverse  functions 108 

96.  Agenda:  Formulae  of  inverse  functions 109 

97.  Logarithmic  forms  of  inverse  cyclic  functions no 

98.  Agenda  :  Examples  in  the  reduction  of  inverse  cyclic 

forms 112 


CHAPTER   V, 

GRAPHICAL  TRANSFORMATIONS. 

XVI.     Orthomorphosis  Upon  the  Sphere. 

99.  Affix,  correspondence,  morphosis 113 

100.  Stereographic  projection 113 

loi.  Transformation  formulae 114 

102.  The  polar  transformation 116 

103.  Agenda:  Properties  of  the  stereographic  transformation  117 


Xll  CONTENTS. 

XVII.     Planar  Orthomorphosis. 

ARTICLE.  PAGE. 

104.  The  Z£/-plane  and  the  ^--plane 117 

105.  The  logarithmic  spirals  of  ^"  non-intersecting  .   ...  118 

106.  Orthomorphosis  of  B"' 119 

107.  Isogonal  relationship 121 

108.  Orthomorphosis  of  cosk  (/^  +  zV) 123 

109.  By  confocal  ellipses 124 

110.  By  confocal  hyperbolas 126 

111.  Agenda:   Problems  in  orthomorphosis 127 


CHAPTER   VI. 
PROPERTIES  OF  POLYNOMIALS. 

xviii.     Roots  of  Complex  Quantities. 

112.  Definition  of  an  «*'^  root 128 

113.  Evaluation  of  «*^  roots 128 

114.  Agenda:   Examples  in  the  determination  of  «*^  roots  .  129 

XIX.     Polynomials  and  Equations. 

115.  Definition  of  a  polynomial 130 

u6.     Roots  of  equations 131 

117.  The  remainder  theorem 131 

118.  Argand's  theorem 133 

119.  Every  algebraic  equation  has  a  root 136 

120.  The  fundamental  theorem  of  algebra 136 

121.  Agenda  :   Theorems  concerning  polynomials 138 


APPENDIX. 
SOME    AMPLIFICATIONS. 

Art.  23,  page  40.    On  the  notation  b^ 139 

Art.  24,  page  42.  On  the  proof  of  the  theorem  :  If  the 
modulus  be  changed  from  7n  to  km  the  corre- 
sponding base  is  changed  from  <^  to  <5'/* 139 

Art.  27,  pages  43,  44.     Alternative  proof  of  the  law  of  indices    139 


SIGNS    AND    ABBREVIATIONS. 

The  following  quantitive  and  operational  signs  are  used  in  this 
work  and  are  collected  here  for  convenience  of  reference. 

+  Plus. 

—  Minus. 

(\i  Difference  between. 

±  Plus  or  minus. 

X  Multiplied  by. 

/  Divided  by. 

y/  Multiplied  or  divided  by. 

=  Is  equal  to. 

=:  Approaches  as  a  limit. 

>>  Is  greater  than. 

<^  Is  less  than. 

^  Is  not  less  than. 

"^  Is  not  greater  than. 

(  >>)  Something  greater  than. 

(<  )  Something  less  than. 

tsr  Tensor  of  (absolute  value  of), 

vsr  Versor  of 

amp  Amplitude  of  (argument  of). 

In  Natural  logarithm  of 

e  Natural  base. 

TT  Ratio  of  circumference  to  diameter, 

''log  Logarithm  of,  to  base  d. 

log^  Logarithm  of,  to  modulus  k, 

•"^    [  Sine,  cosine,  etc.,  to  modulus  k. 
cos,,  etc.)  ' 


"^it  j    Limit  of,  when  :r  approaches  a. 
=  a) 


limit 

.*.     Therefore. 


UNIPLANAR   ALGEBRA 


INTRODUCTION. 


EUCLID'S   DOCTRINE  OF   PROPORTION. 
A. — Notation. 

In  Sections  B  and  C  of  this  Introduction  capital  letters 
denote  magnitudes,  and  when  the  pairs  of  magnitudes  com- 
pared are  both  of  the  same  kind  they  are  denoted  by  letters 
taken  from  the  early  part  of  the  alphabet,  as  A,  B  compared 
with  C,  D ;  but  when  they  are,  or  may  be,  of  different 
kinds,  by  letters  taken  from  different  parts  of  the  alphabet, 
as  A,  B  compared  with  P,  Q  or  X,  Y.  Small  italic  letters 
m,  71,  p,  q  denote  integers.  By  vi .  A  or  vi  A  is  meant  the 
w^^  multiple  of  A,  and  it  may  be  read  in  times  A ;  by  vin 
is  meant  the  arithmetical  product  of  the  integers  in  and  ;/, 
and  it  is  assumed  that  vi  n  =  nm.  The  combination  m .  7ih. 
denotes  the  w"'  multiple  of  the  7^^''  multiple  of  A,  and  it  is 
assumed  that  m .  n\  =  n .  in  A  =  vi  n .  A. 

B. — Definitions  and  Axioms.^'' 

1.  "  A  greater  magnitude  is  said  to  be  a  multiple  of  a 
less,  when  the  greater  contains  the  less  an  exact  number  of 
times. ' ' 

2.  "A  less  magnitude  is  said  to  be  a  submultiple  of  a 
greater,  when  the  less  is  contained  an  exact  number  of  times 
in  the  greater." 


*  The  quoted  paragraphs  of  Section  B  are  transcribed  in  part  from  the  Sylla- 
bus of  Plane  Geometry^  published  by  the  Association  for  the  Improvement  of 
Geometrical  Teaching,  in  part  from  Hall  and  Stevens'  Text  Book  of  EucluVs 
Elements,  Book  V. 


4  INTRODUCTION. 

The  following  property  of  multiples  is  assumed  as 
axiomatic  : 

( i ).  7;2  A  >  =  or  <  w  B  according  as  A  >  =  or  <  B. 
(Euc.  Axioms  i  and  3.) 

The  converse  necessarily  follows  :  ^^ 

(ii).  A  >  =  or  <  B  according  as  m\  >  =  or  <  vi  B. 
(Euc.  Axioms  2  and  4.) 

3.  ' '  The  ratio  of  one  magnitude  to  another  of  the  same 
kind  is  the  relation  which  the  first  bears  to  the  second  in 
respect  of  quantuplicity . ' ' 

"The  ratio  of  A  to  B  is  denoted  thus,  A  :  B  ;  and  A  is 
called  the  antecedent,  B  the  consequent  of  the  ratio. ' ' 

"The  quantuplicity  of  A  with  respect  to  B  may  be 
estimated  by  examining  how  the  multiples  of  A  are 
distributed  among  the  multiples  of  B,  when  both  are  arranged 
in  ascending  order  of  magnitude  and  the  series  of  multiples 
continued  without  limit."  This  distribution  may  be 
represented  graphically  thus  : 


D        A 

2  A 

3  A 

4A 

5A 

6  A 

7  A 

8A 

1    1    1    1    1    1    1    1 
1      1      1      1      1      1 

Multiples  of  A  : 

Multiples  of  B :  ,  „        „        „        „ 

^  o  B  2B         3B         4B         5B         6B 

Fig.  I. 

4.  If,  in  this  comparison  of  the  multiples  of  two 
magnitudes,  any  multiple,  as  uK,  of  the  one  coincides 
with  (is  equal  to)  any  multiple,  as  ;;2B,  of  the  other,  the  two 
magnitudes  bear  the  same  ratio  to  one  another  as  the  two 
numbers   vi,  n,  and    are    said    to    be    commensurable,    but 


*  "  Rule  of  Conversion.  If  the  hypotheses  of  a  group  of  demonstrated 
theorems  be  exhaustive  (that  is,  form  a  set  of  alternatives  of  which  one  must  be 
true),  and  if  the  conclusions  be  mutually  exclusive  (that  is,  be  such  that  no  two  ot 
tham  can  be  true  at  the  same  time),  then  the  converse  of  every  theorem  of  the 
group  will  necessarily  be  true."    {Syllabus  of  Plane  Geometry,  p.  5.) 


INTRODUCTION.  5 

Dicommensiirable    if    no    such    coincidence     takes     place, 
however  far  the  process  of  comparison  is  carried. 

5.  The  ratio  of  two  magnitudes  is  said  to  be  equal  to  a 
second  ratio  of  two  other  magnitudes  (whether  of  the  same 
or  of  a  different  kind  from  the  former),  when  the  multiples 
of  the  antecedent  of  the  first  ratio  are  distributed  among 
those  of  its  consequent  in  the  same  order  as  the  multiples 
of  the  antecedent  of  the  second  ratio  among  those  of  its 
consequent. 

As  tests  of  the  equality  of  two  ratios  either  of  the  follow- 
ing criteria  may  be  employed,  m  and  n  being  integers  : 

(i).     The  ratio  of  A  to  B  is  equal  to  that  of  P  to  O, 
when  ;;^  A  >  =  or  <  7z  B 

according  as  w  P  >  =  or  <  ;^  Q. 

(ii).  If  VI  be  any  integer  whatever  and  n  another  inte- 
ger so  determined  that  either  mh.  is  between  ;^  B  and 
(;^^-I)B  or  is  equal  to  nV>,  then  the  ratio  of  A  to  B  is 
equal  to  that  of  P  to  Q, 

when  rn?  is  between  nQ  and  (?^+  i)Q  or  equal  to  ?^0 

according  as  nik.  is  between  7^B  and  (;z+  i)B  or  equal  to  ;zB. 

(iii).  It  should  be  remarked  that  the  rule  of  identity^  is 
applicable  to  this  definition,  and  that  therefore,  if  the  ratio 
of  A  to  B  be  equal  to  that  of  P  to  Q, 

then  also  w  A  >  =  or  <  7z  B 

according  as  w  P  >  Z3=  or  <  n  O. 

6.  "When  the  ratio  of  A  to  B  is  equal  to  that  of  P  to 
O,  the  four  magnitudes  are  said  to  be  proportionals,  or  to 
form  ^proportion.      The  proportion  is  denoted  thus  : 
A  :  B  ::  P  :  Q, 

*  Rule  of  Idf.ntity.  If  there  be  only  one  fact  or  state  of  things  X,  and  only- 
one  fact  or  state  of  things  Y,  then  from  X  is  Y  the  converse  Y  is  X  of  necessity- 
follows. 


INTRODUCTION. 


3\      4A      5A      6A      7A 

1'     ',    '    ,'     ', 


B  2B  3B  4B  5B 

P        2?       3P       4P       5P       6P       7P 


which  is  read  :  A  is  to  B  as  P  is  to  O.     A  and  O  are  called 
the  extremes,  B  and  P  the  means. ' ' 

The  proportion  A  :  B  : :  P  :  Q  may  be  represented  graph- 
ically thus  : 

Multiples  of  A  : 

Multiples  of  B  : 

Multiples  of  P  :  ,        ,  ,        ,        , 

^                           '    I    '       ,'        I        I        '        ' 
Multiples  of  Q  : 1 1 \ I L 

'^  o  Q  2Q  sQ  4Q  5Q 

Fig.  2. 

In  a  diagram  of  this  kind  it  is  obvious  that  in  general, 
of  the  two  figures  thus  compared,  representing  tw^o  equal 
ratios,  one  will  be  an  enlarged  copy  of  the  other,  but  in 
particular,  if  the  antecedents  and  consequents  be  respec- 
tively equal  to  one  another,  the  two  figures  will  be  con- 
gruent. 

7.  "The  ratio  of  one  magnitude  to  another  is  greater 
than  that  of  a  third  magnitude  to  a  fourth,  when  it  is  possi- 
ble to  find  equimultiples  of  the  antecedents  and  equimulti- 
ples of  the  consequents  such  that  while  the  multiple  of  the 
antecedent  of  the  first  ratio  is  greater  than,  or  equal  to, 
that  of  its  consequent,  the  multiple  of  the  antecedent  of  the 
second  is  not  greater,  or  is  less,  than  that  of  its  consequent." 
That  is, 

A  :  B  >  P  :  O, 

if  integers  m,  n  can  be  found  such  that 

if  m  A  >  72  B,  then  w  P  ^  72  Q, 

or  if  mK  =  n B,  then  772 P  <  72 Q. 

8.  "If  A  is  equal  to  B,  the  ratio  of  A  to  B  is  called  a 
ratio  of  equality. 


INTRODUCTION.  7 

"If  A  is  greater  than  B,  the  ratio  of  A  to  B  is  called  a 
ratio  of  greater  inequality. 

"If  A  is  less  than  B,  the  ratio  of  A  to  B  is  called  a  7'atio 
of  less  inequality. '  * 

9.  "Two  ratios  are  said  to  be  reciprocal  when  the  ante- 
cedent and  consequent  of  one  are  the  consequent  and  ante- 
cedent of  the  other  respectively." 

10.  "Three  magnitudes  of  the  same  kind  are  said  to  be 
proportionals  when  the  ratio  of  the  first  to  the  second  is 
equal  to  that  of  the  second  to  the  third." 

11.  "Three  or  more  magnitudes  are  said  to  be  in  con- 
tinued proportion  when  the  ratio  of  the  first  to  the  second  is 
equal  to  that  of  the  second  to  the  third,  and  the  ratio  of 
the  second  to  the  third  is  equal  to  that  of  the  third  to  the 
fourth,  and  so  on." 

C. —  Paraphrase  from  the  Fifth  Book  of  Euclid.* 
Proposition  i. 

*  ^Ratios  which  are  equal  to  the  same  ratio  ai^e  equal  to 
one  another. ' ' 

Let  A  :  B  : :  P  :  Q  and  X  :  Y  : :  P  :  Q,  then  A  :  B  : :  X  :  Y. 

For,  the  multiples  of  each  of  the  six  magnitudes  being 
ranged  beside  each  other  in  pairs,  as  represented  in  Def.  6, 
it  is  obvious  that  in  each  case  the  multiples  of  the  ante- 
cedent are  distributed  among  those  of  the  corresponding 
consequent  in  exactly  the  same  order. 


*  Enunciations  and  numbering  quoted  from  Hall  and  Stevens'   Text  Book  of 
Euclid's  Elements,  Book  V. 


INTRODUCTION. 


Proposition  2. 


'■'If  two  ratios  be  equal,  the  antecedent  of  the  second  is 
greater  than,  equal  to,  or  less  than,  its  consequent,  according 
as  the  antecedoit  of  the  first  is  greater  than,  equal  to,  or  less 
than,  its  consequent.'' 

Let  A:B::P:0; 

then  P  >  =  or  <  O  according  as  A  >  =  or  <  B. 

This  follows  from  Def  5  (iii)  by  taking  m  =  n  ^=  i. 

Proposition  3. 

''If  two  ratios  be  equal,  their  reciprocal  ratios  are  equal.''' 
(Invertendo.) 

Let  A  :  B  ::  P  :  O,  then  B  :  A  ::  O  :  P. 

This  is  made  evident  graphically  by  constructing  dia- 
grams for  the  two  ratios,  as  in  Def  6,  and  reading  them  off 
as  a  proportion,  once  in  direct  order  (downwards)  for  the 
hypothesis,  and  again  in  inverse  order  (upwards)  for  the 
conclusion,  and  applying  the  rule  of  identity. 

Proposition  4. 

"Equal  magnitudes  have  the  same  ratio  to  the  same  mag- 
7iitude;  and  the  same  magiiitude  has  the  same  ratio  to  equal 
magnitudes. ' ' 

Let  A,  B,  C  be  three  magnitudes  of  the  same  kind,  and 
let  A  =  B  ;  then  A  :  C  ::  B  :  C  and  C  :  A  ::  C  :  B. 

For  if  A  =  B,  their  multiples  are  identical,  and  are 
therefore  distributed  in  the  same  order  among  those  of  C. 

.-.   A  :  C  ::  B  :  C,  (Def  5.). 

and  invertendo  C  :  A  ::  C  :  B.  (Prop.  3.) 


INTRODUCTION.  9 

Proposition  5. 

''Of  two  unequal  magnitudes,  the  greater  has  a  greater 
ratio  to  a  third  niag7iitude  than  the  less  has;  and  the  same 
magnitude  has  a  greater  ratio  to  the  less  of  two  magnitudes 
than  it  has  to  the  greater. ' ' 

Let  A,  B,   C  be   three  magnitudes    of  the  same  kind  ; 

then  if  A  >  B,  so  is  A  :  C  >  B  :  C, 

and  if  B  <  A,  so  is  C  :  B  >  C  :  A. 

First :  If  A  >  B,  an  integer  m  can  be  found  such  that 
m  A  exceeds  7n  B  by  a  magnitude  greater  than  C.  Hence, 
the  integer  7i  being  so  chosen  that  mA  is  equal  to  or 
greater  than  nC  and  less  than  (?2-}-i)C,  the  conditions  re- 
quire that  D  /  ^ 
wB  <  ;^C, 

and  therefore  A  :  C  >  B  :  C.  (Def.  7.) 

Second  :  If  B  <  A,  then  taking  m  and  71  as  in  the  fore- 
going proof  the  same  inequalities  between  the  multiples  of 
A,  B  and  C  still  exist,  and  because  ?^C  ^  ?;2B  while  71Q  is 
either  •<  711 K,  or  at  most  =  fJiA, 

.'.   C  ;  B  >  C  :  A.  (Def  7.) 

Proposition  6, 

''Magnitudes  which  have  the  same  ratio  to  the sa7ne  77iag- 
7iitude  are  equal  to  one  a7iother;  and  those  to  which  the  sa7ne 
7nagnitude  has  the  sa7ne  ratio  are  equal  to  07ie  a7iother. ' ' 

That  is,  A,  B,  C  being  three  magnitudes  of  the  same  kind ; 

if  A  :  C  ::  B  :  C,  then  A  =  B. 

and  if  C  :  A  ::  C  :  B,  then  A  =  B. 

The  proof  of  this  proposition  is  made  a  part  of  the  proof 
oi  Prop.  7. 


lO  INTRODUCTION. 

Proposition  7. 

*  ' '  That  magnitude  which  has  a  greater  ratio  than  another 
has  to  the  same  magnitude  is  the  greater  of  the  two;  and 
that  magnitude  to  which  the  same  has  a  greater  ratio  than  it 
has  to  another  magnitude  is  the  less  of  the  two'' 

That  is,  A,  B,  C  being  three  magnitudes  of  the  same  kind; 
if  A  :  C  >  B  :  C,  then  A  >  B. 

and  if  C  :  A  >  C  :  B,  then  A  <  B. 

Proof  of  Propositions  6  and  7. 

First  Part :    It  has  been  proved  that 

A  :  C  ::  B  :  C,  if  A  =  B,  (Prop.  4.) 

and  A  :  C  >  B  :  C,  if  A  >  B,  (Prop.  5.) 

and  A  :  C  <  B  :  C,  if  A  <  B.  (Prop.  5) 

Hence,  hy  \.\\e  rule  of  conversion,  (Def  2,  Ax.  ii.) 

A  >  =  or  <  B 
according  as  A  :  C  >  =  or  <  B  :  C. 

This  proves  the  first  part  of  each  of  the  two  propositions. 

Second  Part :    It  has  been  proved  that 

C  :  A  ::  C:B,  if  A  =- B,  (Prop.  4.) 

and  C  :  A<  C  :  B,  if  A  >  B,  (Prop.  5.) 

and  C  :  A  >  C  :  B,  if  A  <  B.  (Prop.  5.) 

Hence,  by  the  7'ule  of  conversion,  (Def  2,  Ax.  ii.) 

A  >  =:-  or  <  B 
according  as  C  :  A  <  =  or  >  C  :  B. 

This  proves  the  second  part  of  each  of  the  two  propositions. 

(i).  Corollary  .•  7^  A  :  C  >  B  :  C,  then  C  :  A  <  C  :  B, 
and  conversely. 

(ii).  Corollary  .•  i^  A  :  C  >  P  :  R,  then  C  :  A  <  R  :  P, 
and  conversely. 


INTRODUCTION.  II 

Proposition  8. 

''Magnitudes  have  the  same  ratio  one  to  another  which 
their  equimultiples  have. ' ' 

Let  A,  B  be  two  magnitudes  of  the  same  kind  and  m 
any  integer ;  then 

A  :  B  ::  ;;/A  :  wB. 
For  if/,  q  be  any  two  integers, 

m  ./>  A  >  =  or  <  w .  ^  B 
according  as  /  A  >  ==  or  <  ^B.        (Def  2,  Ax.  ii.) 

But  VI. p  A  =^p.  m  A  and  7n.qB  =  q.7nB; 

.  • .    p.7nA^  =  or<,q.mB 
according  as  /  A  >  =  or  <  ^B, 

whatever  integers  p  and  q  represent.      Hence 
A  :  B  ::  mA  :  niB. 

(i).  Corollary.-  ^  A  :  B  ::  P  :  Q,  then  w  A  :  viB 
::   ?iF  :  ?iQy  whatever  integers  vi  and  n  may  be. 

Proposition  9. 

''If  two  ratios  be  equal,  and  any  equimultiples  of  the 
antecedents  and  also  of  the  co7iseque7its  be  taken,  the  77iulti- 
ple  of  the  first  a7itecede7it  has  to  that  of  its  co7iseque7it  the 
same  ratio  as  the  multiple  of  the  other  a7itecede7it  has  to  that 
of  its  conseque7it.''^ 

Let  A  :  B  ::  P  :  O,  then  ;;^A  :  ;^B  ::  771B  :  ;zO,  w  and 
n  being  any  integers. 

For,  if/,  q  be  any  two  integers,  then,  since  by  hypothesis 
A  :  B  :;  P:  O, 
.  • .  pin.B  >  =  or  <  q7i.O 
according  as        /w.  A  >  =  or  <  ^?^.  B  ;  (Def  5.) 

that  is,  p .  77iB  y>  ~  or  <^  q .  nQ 

according  as        /.  7nA  >  ^  or  <  ^.  72B. 

.'.   ?;^A  :  ;^B  ::  wP  :  ;^0.  (Def  5.) 


12  INTRODUCTION. 


Proposition  io. 


''If four  magnitudes  of  the  same  kind  be  proportio?ials, 
the  first  is  greater  tJian,  equal  to,  or  less  than,  the  third, 
according  as  the  second  is  greater  than,  equal  to,  or  less  tha?i, 
the  fourth. ' ' 


Let 

A  :  B  ::  C  :  D; 

^n    A  >  = 

or  <  C  according  as  B  >  = 

=  or  <  D. 

If  B  >  D, 

then 

A  :  B  <  A  :  D; 

A  :  B  ::  C  :  D, 
.'.  C  :D<  A  :  D, 
.-.   A:D>C  :  D, 

.'.   A>C. 

(Prop.  5.) 
(Hypoth.) 

(Prop.  7.) 

If  B  =  D, 

then 

C:  B  ::  C:  D; 
A:B  ::  C:D, 
.-.   A  :  B  ::  C  :  B, 
.-.   A  =  C. 

(Prop.  4.) 
(Hypoth.) 
(Prop.  I.) 
(Prop.  6.) 

If  B  <  D, 

then 

A  :  B>  A  :  D; 
A  :  B  ::  C  :  D, 
.'.   C  :  D>  A  :  D, 
.-.   C>  Aand  A  <  C. 

(Prop.  5.) 
(Hypoth.) 

(Prop.  7.) 

Proposition  it. 

''If four  magnitudes  of  the  same  kind  be  proportionals ^ 
the  first  will  have  to  the  third  the  same  ratio  as  the  second  to 
the  fourth. ' '     (  Alternando. ) 

Let       A  :  B  ::  C  :  D,  then  will  A  :  C  ::  B  :  D. 


INTRODUCTION. 


13 


For  m  and  n  being  integers, 

A  :  B  ::  niP>^  :  ;;zB 
and  C  :D  ::  /I  C  :  nD, 

.  • .   mA  >  =^  or  <  7iC  according  as  7;^B  > 


But  VI  and  7i  are  any  integers  ; 
.-.   A  :  C  ::  B 


D. 


(Prop.  8.) 
(Prop.  I.) 
or  <  7iD. 
(Prop.  ID.) 

(Def.  5.) 


D. — Seven  Fundamental  Theorems  in  Proportion.* 
Proposition  12:  ( Lemma). ''^'* 
''  If  on  two  straight  lines,  AB,  CD,  cut  by  two  parallel 
straight  lines  A  C,  B  D,  equimultiples  of  the  i?itercepts  re- 
spectively be  taken;  the?i  the  lijie  joining  the  points  of 
division  will  be  parallel  to  AQ^  ^r  B  D. " 

On  AB  and  CD,  produced  either  way,  let  the  respect- 
ive equimultiples  BE,  D  F  of  A  B,  C  D  be  taken,  on  the 
same  side  of  B  D  ;  then  E  F  is  parallel  to  B  D. 

For,  join  A  D,  D  E,  B  C,  B  F.  Since  the 
triangles  A  B  D,  C  B  D  are  on  the  same  base 
BD,  and  their  vertices  A,  C  are  in  the  line 
AC  parallel  to  BD,  they  are  equal  in  area; 
and  whatever  multiple  BE  is  of  A B,  or  DF  of 
CD,  the  triangle  DBE  is  that  same  multiple  of 
the  triangle  ABD,  and  the  triangle  DBF  of 
the  triangle  C  B  D. 

.  ■ .  area  of  triangle  E  B  D  =:  area  of  triangle 
FBD. 

But  these  triangles  E  B  D,   FBD  have  the 
same  base  B  D  ;  hence  their  vertices  E  and  F 
must   be   in   a   straight  line  parallel  to  BD  and  therefore 
EF  is  parallel  to  BD. 

*  Enunciations  of  Propositions  13-17  quoted  from  the   Syllcifjus  of  Plane  GeO' 
metry,  Book  IV,  Section  2. 

**J.  M.  Wilson:   Elementary  Geometry,  page  205. 


Fig.  3 


14 


INTRODUCTION*. 


Proposition  13. 

''  If  two  straight  lines  be  cut  by  three  parallel  straight 
lines,  the  intercepts  cni  the  one  are  to  one  another  in  the  same 
ratio  as  the  correspandijig  iyiiercepts  on  the  other. ' 

Let  the  three  parallel  straight  lines  A  A',  B  B',  CC  be 
cut  by    t\s-o  other  straight  lines  A  C,  A'  C  in  the  points 
A.  B.  C  and  A'.  B',  C  respectively;  then 
AB  :  BC  ::  A' B'  :  B*  C. 

For.  on  A  C  take  B  M  =  m  .  A  B.  B  X  =  ;^  •  B  C.  m  and 
n  being  integers,  M  and  X  on  the  same  side  ot  B.     Also  on 

A'C  takeB'M'  =  ;;^.  A'B',  B' X' =  ;^.  B'C.      _a a^ 

M'  and  X'  being  on  the  same  side  of  B'  as  M 

and  N  of  B.      Then,  by  the  foregoing  lemma      b 5' 

(Prop.  12),  M  M'  and  X  X'  are  both  parallel  to     ^  '^, 

B  B'  and  cannot  meet.     Hence,  whatever  in- 
tegers in  and  n  may  represent, 
B'  M'  (or  7//.  A'  B'  )>  =  or  <  B'X'  (or  n.  B'  C) 

according  as 

B  M  (or  m.  A  B )  >  =  or  <  B  X  (or  n.  B  C); 
.-.   AB  :  BC  ::  A'B'  :  B'C. 

(i).    Corollary  :   '■'  If  tlie  sides  of  a  tri-  ^^r-  -f 

angle  be  cut  by  a  straight  line  parallel  to  the  base,  the  seg- 
yjients  of  one  side  are  to  one  another  in  the  same  ratio  as  the 
segmetits  of  the  other  side. ' ' 

(ii).  Corollary:  '' If  two  straight  lines  be  cut  by 
four  or  more  parallel  straight  lines,  the  ijitercepts  on  the  07ie 
are  to  cme  ayiother  in  the  same  ratio  as  tlie  correspondiyig  iji- 
tercepts on  the  other' ' 

(iii).  Corollary  :  If  in  any  triayigle,  as  O  A  B,  a 
straight  line  E  F,  parallel  to  the  side  A  B.  ait  the  other  sides ^ 
O  A  in  E  and  O  B  in  F,  then 

AB  :  EF  ::  OA  :  OE  ::  OB  :  OF. 


INTRODUCTION. 


15 


Proposition  14. 
''  A  given  finite  straight  line  can  be  divided  internally 
into  segments  having  any  given  ratio,  and  also  externally 
into  segments  having  any  given  ratio  except  the  ratio  of 
equality ;''  and  if  the  line  be  given  in  both  length  and  sense, 
there  is  in  each  case  one  and  only  one  such  point  of  division. 

Let  A  B  be  the  given  straight  Hne ;    it  may  be  divided, 
as  at  E,  in  a  given  ratio  P  :  Q. 

For,  on  the  straight  line  A  G  making  any  convenient 
angle  with  A  B  lay  off  A  C  =  P,  C  D  =  Q.  Then  C  E, 
drawn  parallel  to  D  B  to  meet 
AB  in  E,  will  divide  AB  at  E 
in  the  given  ratio.     (Prop.  13.) 

Since  C  E  and  D  B  are  par- 
allel, C  and  E  lie  on  the  same 
side  of  D  and  B,  and  hence  the 
division  will  be  internal  if  A  and  D 
are  on    opposite    sides  of   C,    but 
external  if  A  and  D  are  on  the  same 
side  of  C. 

If  the  line  to  be  divided  be  esti- 
mated in  a  given  sense,  as  from  A 
to  B,  there  is  in  each  case  only  one 
point  of  division  in  the  given  ratio, 
as  F,  be  joined  to  C  and  B  G  be  drawn    parallel  to  F  C, 
then  AF  :  FB  ::  AC  :  CG,  (Prop.  13.) 

so  that  F  divides  AB  in  the  ratio  AC  :  CG,  different  from 
the  given  ratio. 

If  the  given  ratio  be  a  ratio  of  equality,  the  construction 
in  the  case  of  external  division  fails. 


Fis-  5 


For  if  any  other  point? 


PRorosiTioN  15. 
' '  A  straight  line  which  divides  the  sides  of  a  triangle 
proportionally  is  parallel  to  the  base  of  the  triangle. ' ' 


i6 


INTRODUCTION. 


Let  DE  divide  the  sides  AB,  AC  of  the  triangle  ABC 
proportionally,  so  that 

AD:  DB::  AE  :  EC, 
then  D  E  is  parallel  to  B  C. 

If  possible,   let  DF  be  parallel  to  B  C,    F  some  other 
point  than  E ;  then 

AD:DB::AF:FC.  .A 

But  by  hypothesis  ^   ^°P*   ^^'^ 

AD:DB  ::  AE:EC 
.-.   AF  :FC::  AE  :  EC 
which  is  only  possible  when  F  coincides 
with  E. 


Fig.  6. 


Proposition  i6. 

"■' Rectangles  of  equal  altitude  are  to  one  another  in  the 
same  I'atio  as  their  bases. 

Let  K  A,  K  B  be  two  rectangles  having  the  common  alti- 
tude OK  and  their  bases  OA,  OB  extending  in  the  same 
line  from  O  to  the  right ;  then 

rect.  KA  :  rect.  KB  ::  OA  :  OB. 

In   the   line    O  A  B    pro-   k 
duced  take 

OM-=w.OA,  ON=;^.OB, 
m  and  7i  being  integers,  and 
complete  the  rectangles  KM,  Fig.  7. 

K  N.  Whatever  multiples  O  M  and  O  N  are  of  O  A  and 
OB,  the  rectangles  KM*  and  KN  are  the  same  respective 
multiples  of  the  rectangles  K  A  and  K  B  ;  that  is, 

KM  =  7n.KA,  KN  =  ?z.KB, 


A      B 


M     N 


INTRODUCTION. 


17 


and  according  as 

OM  (or  ;;^.OA)  >  =  or  <  ON  (or  w.OB) 
so  is     KM  (or  w.KA)  >  =  or  <  KN  (or?^.KB) 

.-.   rect.  KA  :  rect.  KB  ::  OA  :  OB.        (Def.  5.) 

(i).    Corollary:    '' Parallelograms  or  triangles  of  the 
same  altitude  ai'e  to  one  another  as  their  bases. 


Proposition  17. 

"/;^  the  same  circle,  or  in  equal  circles,  angles  at  the  cen- 
tre and  sectors  are  to  one  another  as  the  arcs  on  which  they 
stand. ' ' 

Let  there  be  two  equal  circles  with  centres  at  K  and  K', 
and  on  their  circumferences  any  two  arcs  O  A,  O'B  ;  then 

angle  OKA  :  angle  O'K'B  ::  OA  :  O'B, 
and  sector  OKA  :  sector  O'K'B  ::  O A  :  O'B. 

(b) 


On  the  two  circumferences  respectively  take 

OM  =  w.OA,  O'N  =  ?^.0'B, 

m  and  n  being  integers.  Whatever  multiples  OM  and 
O'N  are  of  O  A  and  O'B,  the  same  multiples  respectively 
are  the  angles  or  sectors  O  K  M  and  O'K'N  of  the  angles  or 
sectors  OKA  and  O'K'B  ;  that  is, 

O KM  =:  W.OKA,  O'K'N  =  ;z. O'K'B, 


1 8  INTRODUCTION. 

and  according  as 

OM  (or  ;;^.0  A)  >  =  or  <  O'N  (or  ?z.O'B) 
so  is 
OKM  (or  ?«.OKA)  >  =  or  <  O'K'N  (or  «.0'K'B). 
.-.   OKA  :  O'K'B  :;  OA  :  O'B,  (Def.  5.) 

wherein  OKA  and  O'K'B  represent  either  angles  or  sectors. 

(i).  CoFtOLLARY  :  III  any  tivo  given  concentric  circles, 
corresponding  arcs  intercepted  by  common  radii  bear  always 
the  same  ratio  to  one  ajiother. 

That  is,   if  u,  21' ,  it",  ...   be  arcs  on  one  of  the  circles 

determined  by  a  series  of  radii,  and  the  same  radii  intercept 

on  the  other  circle  the  corresponding  arcs  z\  v' ,  v" ,  .   .   . 

then  ,       ,        .f       „ 

n  \v  v.  2t\v   ::  u    :  V    ... 

Proposition  iS. 

Arcs  of  circles  that  subtend  the  same  angle  or  equal 
angles  at  their  centres  are  to  one  another  as  their  radii. 


c'l 


fe 


A  A  ^  ^ 

Fig:      9. 

Let  there  be  two  arcs  S  ^=  AD  and  S' =  A'D',  havin 
their  angles  AOD  and  A'OD'  either  equal  and  distinct  or 
common,  and  let  R,  R'  be  their  respective  radii ;  then 

S  :  S'  ::  R  :  R'. 

If  the  two  arcs  be  not  concentric,  let  them  be  made  so, 
and  let  their  hounding  radii  be  made  to  coincide.     Then 


INTRODUCTION.  I9 

the  proposition  proved  for  the  concentric  will  also  be  true 
for  the  non-concentric  arcs. 

Conceive  the  angle  at  O  to  be  divided  into  m  equal 
parts,  in  being  any  integer,  by  radii  setting  off  the  arcs  S 
and  S'  into  the  same  number  of  equal  parts,  and  draw  the 
equal  chords  of  the  submultiple  arcs  of  S  and  the  like  equal 
chords  of  the  submultiple  arcs  of  S'.  Let  C  and  C  be  the 
respective  lengths  of  these  chords. 

Then,  since  the  chords  C,  C  cut  off  equal  segments  on 
the  Hues  OA',  OB'  they  are  parallel  (Prop.  12),  and 

C  :  C  ::  R  :  R'.      (Prop.  13,  Cor.  iii.) 
Therefore,  vi  being  any  integer, 

mC  :  ;;^C'::  R  :  R'.  (Prop.  8.) 

Let  711  be  the  number  of  equal  parts  into  which  the  angle 
at  O  is  divided  ;  then  in  C  and  vi  C  are  the  lengths  of  the 
polygonal  lines  formed  by  the  equal  chords  of  S  and  S' 
respectively. 

If  now  VI  be  increased  indefinitely,  the  chords  decrease 
in  length  but  increase  in  number,  and  the  two  polygonal 
lines  which  they  form  approach  coincidence  with  the  arcs  S 
and  S'  respectively ;  and  by  increasing  in  sufficiently  the 
aggregate  of  all  the  spaces  between  the  arcs  and  their 
chords  may  be  made  smaller  than  any  previously  assigned 
arbitrarily  small  magnitude.  Under  these  circumstances  it 
is  assumed  as  axiomatic  that  the  relation  existing  between 
the  polygonal  lines  exists  also  between  the  arcs,  which  are 
called  limits.     Under  this  assumption  it  follows  that 

S  :  S'  ::  R  :  R'.  Q.  E.  D. 

(i).  Corollary  :  Circumferences  are  to  one  another 
as  their  radii. 

(ii).  Corollary  :  Of  two  arcs  of  circles  that  subtend 
the  same  angle  or  equal  angles  at  their  centres,  that  is  the 
longer  which  has  the  longer  radius.      (  By  Prop.  2.) 


20  INTRODUCTION. 

E. — Agenda  :  Supplementary  Propositions. 

(i).    If  two  geometrical  magnitudes  A,  B,  have  the  same 
ratio  as  two  integers  vi,  n,  prove  that 
nK  =  m  B. 

(2).  If  A,  B  be  two  geometrical  magnitudes  and  m,  71 
two  integers  such  that  7^  A  =  7«B,  prove  that 

A  :  B  ::  w  :  n. 
Hence  infer  the  statement  in  the  first  part  of  Definition  4, 
page  4,  concerning  commensurable  magnitudes. 

(3).    Given  A  :  B  ::  P  :  Q  and  ;zA  =  ;/^B,   prove  that 
nY  =  mO. 

(4).  It  is  a  corollary  of  (3),  that  if  A  :  B  ::  P  :  O  and 
A  be  a  multiple,  part,  or  multiple  of  a  part  of  B,  then  P  is 
the  same  multiple,  part,  or  multiple  of  a  part  of  O. 

(5).    Given  A  :  B  ::  P  :  Q  and  B  :  C  ::  Q  :  R,  prove  that 
P  >  =  or  <  R  according  as  A  >  =  or  <  C. 

(6).    Given  A  :  B  ::  P  :  O  and  B  :  C  ::  O  :  R,  prove  that 
A  :  C  ::  P  :  R.  (Ex  aequali.) 

(7).  Given  A  :  B  ::  P  :  O  and  B  :  C  ::  O  :  R  and  C  :  D 
::  R  :  S  and  D  :  E  ::  S  :  T,  prove  that 

A  :  E  ::  P  :  T. 
State  and  prove  the  general  theorem  of  which  this  is  a  par- 
ticular case. 

(8).    Given  A  :  B  ::  O  :  R  and  B  :  C  ::  P  :  Q,  prove  that 
A  :  C  ::  P:  R. 


CHAPTER    I. 

LAWS  OF  ALGEBRAIC   OPERATION. 


I.     Quantity. 

I.  Quantities  in  GeneraL  Quantities,  whatever  their 
nature,  may  be  expressed  in  terms  of  geometrical  magni- 
tudes ;  in  particular  they  may  be  thought  of  as  straight 
lines  of  definite  fixed  or  variable  length.  Such  mag- 
nitudes, in  so  far  as  they  represent  the  quantities  of 
ordinary  algebra,  are  of  three  kinds  :  real,  imaginary,  and 
complex  ;  real  if,  when  considered  by  themselves  (laid  off 
upon  the  real  axis),  they  are  supposed  to  involve  only  the 
idea  of  length,  positive  or  negative,  without  regard  to 
direction,  imaginary  when  they  involve  not  only  length, 
but  also  turning  or  rotation  through  a  right  angle,  that  is, 
length  and  direction  at  right  angles  to  the  axis  of  real 
quantities,  finally  complex  if  they  embody  length  and 
rotation  through  any  angle,  that  is,  length  and  unrestricted 
direction  in  the  plane. 

If  we  think  of  the  straight  line  as  generated  by  the 
motion  of  a  point,  we  may  translate  length  positive  or 
negative  into  motion  forwards  or  backwards ;  and  it  will 
sometimes  be  convenient  to  use  the  latter  terminology  in 
place  of  the  former. 

It  is  at  once  evident  that  both  reals  and  imaginaries 
are  particular  forms  of  complex  quantities,  reals  involving 
motion  forwards  or  backwards  and  rotation  through  a 
zero-angle,     imaginaries     involving    motion    forwards    or 


22  LAWS    OF   ALGEBRAIC    OPERATION. 

backwards  and  rotation  through  a  right  angle.  The 
three  kinds  of  quantities  will  be  considered  in  order;  the 
distinction  between  them,  here  roughly  outlined,  will  be 
made  clearer  by  a  study  of  their  properties. 

2.  Nature  of  Real  Quantities.  It  is  evident  that  all 
real  quantities  may  be  made  concretely  cognizable  by  laying 
them  off  (in  the  imagination)  as  lengths,  in  the  positive  or 
negative  sense,  upon  one  straight  line.  In  this  represen- 
tation every  straight  line  suffices  to  embody  in  itself  all  real 
quantities,  having  its  own  positive  and  negative  sense,  that 
is,  its  direction  forwards  and  backwards.  In  particular, 
all  the  numbers  of  common  arithmetic,  both  integral  and 
fractional,  are  accurately  represented  by  distances  laid  off 
from  a  fixed  origin  in  the  positive  sense  upon  a  straight 
line,  and  in  the  same  way  all  so-called  irrational  numbers, 
though  only  approximately  realizable  as  true  numbers  in 
arithmetic,  are  accurately  represented.  Hence  the  following 
proposition,  which  is  postulated  as  self-evident: 

The  laws  of  algebraic  operatio7i  that  obtain  with  geo- 
metrical real  magnitudes,  that  is,  lengths  laid  off  upon 
a  straight  line,  are  ipso  facto  trne  when  applied  to 
arithmetical  quantities,   or  Jimnbers. 

But  the  converse  of  this  proposition  is  not  equally  self- 
evident.  For  inasmuch  as  so-called  irrational  number, 
that  is,  quantity  in  general,  is  not  realizable  as  true  number 
(integer  or  fraction)  in  arithmetic,  the  proof  that  the  laws 
of  algebraic  operation  obtain  for  integers  and  fractions 
constitutes  not  a  proof,  but  only  a  presumption,  that  they 
obtain  also  for  so-called  irrational  number. 

hi  the  following  pages  viagnitudes  will  be  represented  by 
straight  lines  of  finite  length. 


LAWS    OF   ALGEBRAIC    OPERATION.  23 

II.     Definitions  of  Algebraic  Operations. 

3.  Algebraic  Addition.  Simple  addition  is  here 
defined  as  the  putting  together,  end  to  end,  different  hne- 
segments,  or  Hnks,  in  such  a  way  as  to  form  a  one  dimen- 
sional continuum,  that  is,  a  continuous  straight  line.  This 
kind  of  addition  corresponds  to  the  addition  of  positive 
numbers  in  arithmetic. 

Algebraic  addition  takes  account  of  negative  magni- 
tudes, that  is,  of  lines  taken  in  the  negative  sense  (from 
right  to  left,  if  positive  lines  extend  from  left  to  right),  and 
to  add  to  any  line-segment  a  negative  magnitude  is  to  cut 
off  from  its  positive  extremity  a  portion  equal  in  length  to 
the  negative  magnitude.  This  kind  of  addition  includes 
the  addition  and  subtraction  of  positive  numbers  in  arith- 
metic and  introduces  the  new  rule  that  larger  positive 
magnitudes  may  be  subtracted  from  smaller,  producing 
thereby  negative  magnitudes.  We  then  extend  the  idea 
of  negativeness  also  to  number  and  produce  negati\'e 
number,  prefixing  the  sign  —  to  positive  number  as  a 
mark  of  the  new  quality. 

The  result  of  adding  together  algebraically  several 
magnitudes  is  called  a  siun.  In  a  sum  the  constituent 
parts  are  terms. 

4.  Zero  is  defined  as  the  sum  of  a  positive  and  an 
equally  large  negative  magnitude ;  in  symbols, 

-j-  a  —  a  =  o. 

It  is  not  a  magnitude  but  indicates  the  absence  of 
magnitude. 

5.  Algebraic  Multiplication.  On  two  straight  lines 
making  any  convenient  angle  with  one  another  at  O, 
Fig.  10,  lay  off  OA  =  a,  OB  =  d,  and  on  OA  in  the  same 


24  LAWS    OF    ALGEBRAIC    OPERATION. 

direction  as  OA  lay  off  0/=j\  which  shall  be  of  fixed 
length  in  all  constructions  belonging  to  algebra  and  shall 
be  called  the  rea/  2init.     Join  J  and  B  and  draw  from  A 

a  straight  line  parallel  to 
JB  to  intersect  OB  in  M. 
Then  by  Proposition  13 
(p.  14)  the  intercepts  on 
OA,  OM  by  the  parallels 
JB,  AM  are  proportionals, 
and  if  OM=vi, 
J  :  a  \:  b  :  vi. 

The  length  w,  thus  determined,  is  defined  as  the 
algebraic  prodtict,  or  simply  the  product,  of  the  real 
magnitude  a  by  the  real  magnitude  b,  and  is  denoted  by 
ay^b,  Q,x\^y  a-b,  or  more  simply  still  hy  ab  r^  In  a  pro- 
duct the  constituent  parts  2X^  factors. 

The  product  aV^  b  may  also  be  a  factor  in  another  pro- 
duct, consisting  therefore  of  three  factors,  as  (<3;  X  ^)  X  c. 
and  this  may  in  turn  be  a  factor  in  a  product  of  four 
factors,  and  so  on. 


Fig.  10. 


6.    Reciprocals. 
J. 


If  m  be  equal  to  j\   then 
J  :a::b:j 
and  b,  a,    are  called    the  7'e- 
ciprocals  of  a,  b,  respectively. 
These  reciprocals  are  written 
thus: 

j  I  a=^  reciprocal  oi  a, 
j  I  b  =  reciprocal  of  b, 
etc.;  or  more  simply,  since/ remains  unchanged  through- 
out  all    algebraic    operations,    they   may   be    conveniently 


*  Descartes:    la  Geometrie,  reprint  of  iSS6,  p.  2. 


LAWS    OF    ALGEBRAIC    OPERATION.  25 

represented  hy  I  a,  j  d,  etc.      Since  the  means,  in  any  pro- 
portion between  like  magnitudes,  may  be  interchanged, 

i^  b  =  I  a,  then  also  a=^  j  d, 
and  vice  versa. 

A  reciprocal,  being  itself  a  line-magnitude,  may  enter  a 
product  as  one  of  its  factors. 

7.  Idemfactor:  Real  Unit.  If  in  the  proportion 
j  \a\\b  \vi  we  write  a  =j\  that  is,  make  AM  in  Fig.  lo 
coincide  with  JB,  then 

j  \j  \\  b  \  w,  that  is,  m  =  b.  (Prop.  2.) 

But  by  the  definition  of  a  product  m  =j  X  b ; 

.■.jXb=b, 
and  in  particular 

An  operator  which,  Hkey,  as  factor  in  a  product  leaves 
the  other  part  of  the  product  unchanged,  is  called  an 
ide^nf actor. ^^'  This  particular  real  idemfactor  is  what  was 
defined  in  Art.  5  as  the  real  unit.  In  arithmetic  it  is 
denoted  by  the  numerical  symbol   i. 

8.  Quotient.  The  product  defined  by  the  propor- 
tion j  :c  :\  I  a  -.m  is  cX  I  a  and  is  called  the  quotient  of 
c  by  a.  The  sign  X  before  /  may  be  omitted  without 
ambiguity  and  this  quotient  be  denoted  by  the  simpler 
notation  c  \  a,  in  which  c  is  called  the  dividejid  and  a  the 
divisor. 

The  proportion  y  -.awb-.j  defines 

b  =  I  a,  <2  =  /  b,  and  a  y^  b  ■=^j  ; 
hence 

^X  b  ^=^  ayi  I  a  =  a  I  a  =y, 

*  Benjamin  Peirce:  Linear  Associative  Algebra  (1870),  p.  16,  or  American 
Journal  of  Mathematics,  Vol.  IV  (1881),  p.  104. 


26  LAWS    OF    ALGEBRAIC    OPERATION. 

and  so  for  any  magnitude  whatever.  Hence  we  may  de- 
scribe the  real  unit  as  the  quotient  of  any  real  magnitude 
by  itself 

The  quotient  c ;  a  is  also  represented  by  r  -f-  <^,  or  by  -. 
The  latter  notation  will  be  frequently  employed  in  the 
sequel. 

g.     Agenda.     Problems  in  Construction. 

(i).  From  the  definitions  of  Arts.  5,  6  and  8  prove 
that  the  following  construction  for  the  quotient  a  j  b  is 
correct:  On  one  of  two  straight  lines,  making  any  con- 
venient angle  with  one  another,  lay  off  OA  =  a,  OB  =  b, 
on  the  other  OJ=^j\  join  B  and  yand  draw  AM  parallel 
to  B/to  intersect  0/m  M.      OM\s  the  quotient  sought. 

(2).    Given  a  X  a  =  vi,  construct  a. 

(3).    Given  a,  b  and  c,  construct  {a  y^  b)  ^  c. 

(4).  Prove  that  «X^i3>  =or<<^  according  as 
<3^  is  >  =  or  <  I 

(5).  Draw  OX  and  O  Kmaking  any  convenient  angle 
with  each  other  ;  on  O  Flay  off  OJ=j\  OA  =  a,  OB=  b, 
and  on  OJ^take  OJ^^=^j.  A  straight  line  through  /par- 
allel to  OX  will  be  cut  by  J^A  and  J^B  in  two  points  P 
and  Q.  Show  that  if  ^  and  B  are  on  the  same  side  of  (9, 
the  distance  between  P  and  Q  is  PQ  =  j  a  ^  j  b,  where 
CO  means  difference  between,  but  if  A  and  B  are  on  oppo- 
site sides  of  O,  then  PQ  =^  j  a-\-  j  b.  In  this  construction 
a  and  b  are  supposed  to  be  positive  magnitudes. 

10.  Infinity  is  defined  as  the  reciprocal  of  zero;  in 
symbols 

/o  =  CO  . 


LAWS    OF    ALGEBRAIC    OPERATION.  27 

When  a  magnitude  decreases  and  becomes  zero,  its  recip- 
rocal obviously  increases  and  becomes  infinite.  Since  zero 
is  not  a  magnitude,  neither  is  infinity  as  here  defined. 

In  Art.  6  it  was  shown  that  d  =  j  a  implies  also  a  =  j  b] 
hence,  fi-om  the  definition  /  o  =  co  follows 

/  CO    =  o. 

The  construction  for  a  product  (Art.  5)  shows  that  when 
one  of  its  factors  becomes  o  or  co  ,  the  other  remaininsf 
finite,  the  product  itself  is  also  o  or  co  ,  so  that  for  all  finite 
values  oi  a 

a/ao  =  a  y(  0  =  0. 

From  the  definition  of  addition  (Art.  3)  it  is  also 
obvious  that 

±:  o  ±  a  =  ±  a, 

±00  ih  <^  =  dz  CO  . 

II.  Indeterminate  Algebraic  Forms.  When  a  sum 
or  product  assumes  one  of  the  forms  -|-  co  —  co  ,  o  X  co  ,  0/0, 
CO  /  CO  ,  it  is  said  to  be  indeterminate,  by  which  is  meant : 
the  form  by  itself  gives  no  information  concerning  its  own 
value. 

(i).  The  form  -|-  00  —  co  .  On  a  straight  line  ABP 
take  at  random  two  points  A,  B,  so  that  AB  is  any  real  finite 
magnitude  whatever.     Take  P,  Q,  R,  on  the  same  line, 

A  Q  B  R  P 


a  b 

Fig.  12. 

such  that 

AP=  I  AQ  =  la,     BP=I  BR  =  \b, 

and  let  Q  pass  into  coincidence  with  A.      P  then  passes  out 


28  LAWS    OF   ALGEBRAIC    OPERATION. 

of  finite  range,  R  passes  into  coincidence  with  B  and  the 
difference 

AB=    a-  :  b, 

whatever  its  original  value,  assumes  the  form 

/  O  —  /o=CO    —   CO. 

Hence,  taken  by  itself,    :o  —  co  gives  no  information  con- 
cerning its  own  value  and  is  indeterminate. 

(ii).  The  forms  oX'20,0/0,  oo/co.  In  the  figure 
of  Art.  5,  let  tI/ and  y  remain  fixed,  while  MA  and/B,  being 
always  parallel  to  one  another,  turn  about  J/ and  y  until 
JfA  coincides  with,  and  /B  becomes  parallel  to  OM. 
At  this  instant  a  becomes  zero,  d  infinite,  and  aV^  b 
assumes  the  form  o  X  co  ;  and  because  the  original  value 
of  ^  X  ^  is  anything  we  choose  to  make  it,  the  expression 
o  X  00  gives  no  information  concerning  its  own  value  and 
is  therefore  an  indeterminate  form. 

Since  /  o  =  00  and  /  co  =  o,  we  may  replace  o  /  o  by 
o  X  00  and  co  /  00  by  co  X  o.  The  two  forms  0/0  and 
CO  /  00   are  therefore  also  indeterminate. 

An  expression,  such  as  j  a  —  j  b,  that  gives  rise  to  an 
indeterminate  form,  may  nevertheless  approach  a  deter- 
minate value  as  it  nears  its  critical  stage.  To  find  this 
value  is  described  as  evaluating  the  indeterminate  form. 
(See  Arts.  43  and  52.) 

III.     Law  of  Signs  for  Real  Quantities. 

12.  In  Addition  and  Subtraction.  The  sign  +,  by 
definition,  indicates  that  the  magnitude  following  it  is 
to  be  added  algebraically  to  what  precedes,  without 
having  its  character  as  a  magnitude  in  any  way  changed ; 
and  the  sign  —  indicates  that  the  magnitude  immediately 


LAWS    OF    ALGEBRAIC    OPERATION,  29 

following  it  is  to  be  reversed  in  sense  (taken  in  the 
opposite  direction)  and  then  added  algebraically  to  what 
precedes. 

Any  symbolic  representative  of  quantity,  a  letter  for 
example,  unattended  by  either  of  the  signs  +  or—,  but 
still  thought  of  as  part  of  an  algebraic  sum,  is  supposed 
to  have  the  same  relation  and  effect  in  such  a  sum  as  if  it 
had  before  it  the  sign  -[-.  This  usage  necessitates  the 
following  law  of  signs  in  addition : 


+  +  =  -!-, 

H —  =  — , 

=  +> 

- +  =  -; 

+  (+  ^)  =  +  ^, 

+  (-  ^)  = 

—  (—  a)  =  +  a, 

-(+«)  = 

thus 

+  (+^)  =  +  ^,      +(-«)  =  - 

a, 

and  by  a,  unattended  by  any  sign,  is  understood  +  <^. 

13.  In  Multiplication  and  Division.  In  Art.  8  it 
was  agreed  that  aid  shall  stand  for  the  product <a;  X  /^. 
This  convention  requires  that  the  combination  X  /  shall 
produce  /.  Looked  at  from  another  point  of  view,  the 
symbol  X  indicates  that  the  letter  following  it  is  to  be 
used  as  a  factor  with  its  character  as  a  magnitude  un- 
changed, while  /  gives  notice  that  the  reciprocal  of  the 
magnitude  immediately  following  it  is  to  be  used  as  a 
factor. 

Any  symbolic  representative  of  quantity,  a  letter  for 
example,  unattended  by  either  of  the  signs  X  or  /,  but 
still  thought  of  as  part  of  (factor  in)  an  algebraic  pro- 
duct, is  supposed  to  have  the  same  relation  and  effect  in 
such  product  as  if  it  had  before  it  the  sign  X.  The  usage 
here  described  necessitates  the  following  law  of  signs  for 
X   and  /: 


30  LAWS    OF   ALGEBRAIC    OPERATION. 


thus 


X  X  =  X,  X  /=--/, 

//  =  X,  /X=/; 

X(Xa)=Xa,  X  {^  a)  =  I  a, 

/(/a)=^  X  a,  /  (X  a)  =  l  a. 


In  practice  the  sign  X  is  usually  omitted,  or  replaced  by  a 
dot ;  thus  :  a  X  b  ^^  ab  ^=  a-  b. 

14.     In  Combination  with  each  other:   -f,  —  with 

X,  /.  In  the  construction  of  a  product  any  factor  affected 
with  the  negative  sign  —  must  be  laid  off  in  the  sense  opposite 
to  the  one  it  takes  when  affected  with  the  positive  sign  -j-, 
and  the  constructions  involving  negative  factors  lead  to  the 
following  rule : 

An  odd  number  of  negative  factors  produces  a  negative 
product; 

An  even  7nunber  of  negative  factors  produces  a  positive 
product. 

For,  suppose  one  factor,  as  a,  to  be  affected  with  the 
negative  sign.  The  construction  of  ( —  a)  Xbi?,  then  as 
follows :  Lay  off  b  in  the  positive  sense,  say  to  the  right, 
and  Of^  the  real  unit,  in  the  positive  sense,  say  upwards : 

then  —  a  must  extend  down- 
wards along  fO  produced. 
Join  JB  and  draw  AM  par- 
allel to  Bf\o  intersect  BO, 
produced  backwards,  in  M. 
The  product  ( —  a')  yi  b  is 
thus  the  negative  magni- 
Fig.  /J.         '  tude  —  m. 

When  the  product  is  in  the  form  a  X  (—  b),  —  b  must 
be  laid  off  in  the  negative  sense  towards  Jl/,  —  b=  OB, ,  a  in 


A 


LAWS    OF   ALGEBRAIC    OPERATION.  3 1 

the  positive  sense  towards  A^,  a  =  OA^,  and  it  is  easy  to 
prove  by  proportion  in  the  similar  triangles  thus  formed, 
that  the  line  through  A^  parallel  to  JB^  intersects  OB^  in  M. 
It  is  also  obvious  that  -f  «  X  <^  =  -{-  m.  Hence 
(—  «)  X  <5  =  <^  X  (—  <^)  =  —  «  X  ^. 
If  both  factors  are  affected  with  the  negative  sign,  the 
construction  is  as  follows :  Draw  OB  =  —  bm  the  negative, 
J  J  OA    =   —a    in   the   negative, 

OJ=^j  in  the  positive  sense, 
M  AM  parallel  to  B/,  intersect- 
ing BO  produced  in  M.  Then 
0M=  in  is  positive  and  by 
writing  the  proportions  for  the 
similar  triangles  OBJ 3.nd  OMA 
it  is  easy  to  show  that 
«  )  X  (  —  ^  )  =  -f-  ^  X  ^. 
Thus  the  product  of  one  negative  factor  and  any  number 
of  positive  factors  is  negative,  while  every  pair  of  negative 
factors  yields  only  positive  products.  Hence  the  proposi- 
tion, Q.  E.  D. 

IV.     Associative  Law  for  Real  Quantities. 

15.  In  Addition  and  Subtraction.  The  sequence 
of  the  terms  of  a  sum  remaining  unchanged,  the  terms  may 
be  added  separately,  or  in  groups  of  two  or  more,  indis- 
criminately, without  disturbing  the  value  of  the  sum  ;  that  is, 

where  a,  b,  c  may  represent  positive  or  negative  magnitudes 
indiscriminately.  This  is  made  evident  at  once  by  laying 
off  and  comparing  with  one  another  the  lines  a  -\-  b,  c,  and 
a,b  ^  c,  and  a,  b,  c,  taken  in  the  proper  sense  and  in  the 
order  indicated  in  the  three  groupings. 


32  LAWS    OF    ALGEBRAIC    OPERATION. 

In  introducing  negative  magnitudes  the  law  of  combina- 
tions of  the  signs  -f  and  — ,  as  described  in  Art.  12,  must 
be  observed.  The  complete  symbolical  statement  of  the 
associative  law  for  addition  and  subtraction  is  contained  in 
the  formula : 

±  {  ±  a  ±  b)  =  ±1  {±  a)  ±.  {±  b), 

wherein  the  order  of  occurrence  of  the  signs   -f-  and   — 
must  be  the  same  in  the  two  members  of  the  equation. 

16.  In  Multiplication  and  Division.  Construct 
upon  O  A  and  O  M,  Fig.  15,  the  products 

OL  =  aXb,      0M=  a  X  b,      OE^b  X  c, 
ON={ay.  b)  X  c,      ON,  =  a  X  (^  X  c). 

In  this  construction  the  lines  markedy*i,  y"^,  f^,  through  Z,  A'' 
and  By  E  and  J,  C  are,  by  the  rule  for  constructing  products 


E  Bx      N   Ni 

bxc 

f^g-  15- 


ay.b 


(Art.  5),  parallel  to  each  other,  as  are  also  the  lines  A, ,  /i. 
through  A,  N^  and  J,  E,  and  the  lines  k^ ,  k^  through  A,  M 
and  /,  B^.  Hence  by  proportion,  as  determined  by  the 
parallels  /= ,  /i , 

b  :  a  X  b  ::  b  X  c  :  ( a  X  b)  X  c,      (Prop.  13.) 


LAWS    OF    ALGEBRAIC    OPERATION.  33 

and  as  determined  by  the  parallels  k.^,  k^, 

j  \a\'.b\ay^b,  (Prop.  13.) 

.-.    y:^::<^X^:(^X^)  X^.  (Prop,  i.) 

But  the  parallels  /z.,  h^  determine  the  proportion 

j  \a\\by^c\ay^  (^byc). 

Therefore  N,  N^  are  one  and  the  same  point  (Prop.  6)  and 

X  (X^X^)  X^=X«X  (X-^XO- 
Thus  the  sign  X  is  distributive  over  the  successive  factors 
of  a  product;  that  is,  as  here  follows  when  a=^j\ 

x(^xO  =  (x^)x(xo. 

The  same  is  true  of  the  sign  /;  for  the  product  of  any 
magnitude  by  Its  reciprocal  is/  (Art.  6),  and  two  products 
that  have  their  factors  respectively  equal  are  obviously 
themselves  equal  (Art.  5) ;  that  is, 

laya=j,   bylb=j\ 
and  {laya)yibylb)=jyj=J.        (Art.  7.) 

Whence,  by  the  associative  law  in  multiplication  just  proved, 
and  because  the  means  in  a  proportion  between  like  magni- 
tudes may  be  interchanged  (Prop.  11), 

jay  {ayb')y  I  b=j\ 
and  il ay  I b)y{ay  b)  =j; 

.-.     /<iayb)  =  /ay/b.  (Art.  6.) 

Thus  the  associative  law  for  the  sign  /  is  the  same  as  that 
for  the  sign  X ,  and  its  general  statement  is 

wherein  the  order  of  occurrence  of  the  signs  X  and  /  in  the 
two  members  must  be  the  same,  and  the  law  of  their  com- 
bination (Art.  13)  must  be  observed. 


34  LAWS  OF  ALGEBRAIC  OPERATION. 

V.     Commutative  Law  for  Real  Quantities. 

17.  In  Addition  and  Subtraction.  Let  a  and  b 
represent  any  two  lengths  taken  in  the  same  sense  along  a 
straight  line  such  that  a  =  OA  and  b  =  AB ;  then 

a-\-b=  OB. 

o A b; B 

Fig.  16. 

Let  B'  be  so  taken  that  0B'=  b ;  then  B' B  =  a.    Therefore 

a-\-b  =  OB=b-\-a. 

KAB  be  negative  and  equal  to  ~b,  lay  off  a  from  O 
to  the  right,  b  from  the  extremity  of  a  to  the  left.  B  will 
fall  to  the  right  or  left  of  O  according  as  b  is  less  than  or 

B^  O  B  A  B  B  O  A 

■F's:-  17- 

greater  than  a ;  and  in  either  case 

a  —  b=  OB. 

\\\  the  former  OB  will  be  positive,  in  the  latter  negative.      If 
now  B'  be  so  taken  that  B'  0  =  b,  then  B' B  =  a.      Hence 

0B=  —  b-\-a  =  a  —  b. 

lfa±b  =  c  and  d  be  any  fourth  magnitude,  then 

c±d=  :Jzd-{-c  =  aztb±d=  ±  d  ±  b -j- a. 

Obviously  any  algebraic  sum  may  have  its  terms  commuted 
in  like  manner.     Stated  symbolically,  this  law  is 

zh(Z±:b=zt:b±a, 

wherein  each  letter  carries  with  it  like  signs  on  both  sides 
of  the  equation. 


LAWS    OF   ALGEBRAIC    OPERATION. 


35 


i8.  In  Multiplication  and  Division.  On  two  straight 
lines,  meeting  at  a  convenient  angle,  construct  OB,  OB^^ 
each  equal  to  b,  and  OA^,  OA,  each  equal  to  a.  On  OB^ 
take  OJ  =  j\  the  real  unit,  join  JA^^  JB,  draw  AM, 
B^N,  parallel  respectively  to  JB,  JA^,  intersecting  OB 
(produced,  if  necessary)  in  M  and  N,  and  let  OM  =  vi, 
ON  =  11.     By  definition  of  a  product  (Art.  5), 

and  b  y^  a=^n. 

But  the  proportion 

determined  by  the 

parallels  h  and  k  is 

J  :  a  ::  b  :  in. 

b  B  N    nT  (Prop.  13.) 

Fig.  18. 

and  that  determined  by  the  parallels  f  and  g  is 

j  \b'.\a\n, 

which  by  interchange  of  means  (Prop.  11)  becomes 

j  \a  \'.b  \n. 

.'.     in  =  n.  (Prop.  6.) 

M  and  N  are  hence  one  and  the  same  point,  and 

bXa  =  aX  b. 

Inasmuch  as  the  reciprocal  of  a  line-magnitude  is  itself 

a  line-magnitude,  it  follows  from  the  above  demonstration 

that  J        ,  I  r 

by  j  a  =  I  aXb; 

or,  observing  the  law  of  signs  (Art.  13),  this  is 

b  I  a=^  I  a  y  b  ; 

and  if  b  carry  with  it  its  proper  factor  sign,  the  more  expHcit 
form  of  statement  contained  in  this  equation  is 
y  b  I  a=  I  ay  b. 


36  LAWS   OF   ALGEBRAIC    OPERATION. 

The  formula  for  the  commutative  law  in  multiplication  and 
division  may  therefore  be  written 

and  this  is  easily  extended  to  products  of  three  or  more 
factors.  In  this  formula,  the  same  sign  attaches  to  a  or  (^ 
on  both  sides  of  the  equation. 

19.     Agenda:     Theorems  in  Proportion. 

(i).  Prove  that  in  a  proportion  the  product  of  the 
means  is  equal  to  the  product  of  the  extremes. 

(2).  Prove  that  rectangles  are  to  one  another  in  the 
same  ratio  as  the  products  of  their  bases  by  their  altitudes. 
Show  that  the  same  relation  holds  for  parallelograms  and 
triangles. 

(3).  If  a  square  whose  side  is  i  be  taken  as  the  unit 
of  area,  the  area  of  any  rectangle  (or  parallelogram)  is 
equal  to  the  product  of  its  base  by  its  altitude. 


VI.     The  Distributive  Law  for  Real  Quantities. 

20.     With   the    Sign   of    Multiplication.     On   OH 

and  OK  (Fig.    19),  take  OC^c,  OE  =  aA~b,  and  con- 
struct the  product  («+^)X<^-'=  OG.     Then   if  OJ  be 

the  real  unit,  laid  off 
on  OK,  EG  and  JC 
are  parallel  by  con- 
struction. On  OK 
take  OA  =  a,  and 
construct  the  product 
^  X  ^  =  OF.  Then 
_H  AF  and  EG,  being 
parallel  to  JC,  are 
parallel  to  one  another, 
and  FB  drawn  parallel  to  OK  is  equal  to  AE,  that  is, 
FB  =  b,  and  therefore  EG  is  the  product  b  X  <^.      Hence 


LAWS    OF    ALGEBRAIC    OPERATION.  37 

{a-^b)  X  c  =  a  X  c  -^  b  X  c  =  c  X  (^a  ^  b), 

which  is  the  law  of  distribution  in  multiplication  and  addi- 
tion. 

The  construction  and  proof  are  not  essentially  different 
when  some  or  all  the  magnitudes  involved  are  affected  with 
the  negative  instead  of  the  positive  sign. 

The  second  factor  may  also  consist  of  two  or  more 
terms ;  for 

{^-\-b)X{c-^d)=:aX{c^d)-\-bX  {c^d), 

by  the  distributive  law, 

=  (^+^)  Xa-^{c-^d)Xb, 

by  the  commutative  law, 
and  again,  by  the  distributive  and  commutative  laws, 

{c-\-d)Xci^{c-^d)X^l^  =  cXa-^dXa-^cXl^-\-dXb 

=-aXc-\-aXd-\-bXc-VbXd. 
Hence,  replacing  the  sign  +  by  the  double  sign  di,  the 
completed  formula  for  the  distributive  law  in  m.ultiplica- 
tion  is  • 

+  (±:^)X(=t=0+(±^)X(d=^). 

21.  With  the  Sign  of  Reciprocation.  Since^,  ^, 
c,  d,  in  the  formula  last  written,  are  any  real  magnitudes 
whatever,  they  may  be  replaced  by  their  reciprocals,  I  a,  j  b, 
I  c,  /  d.     For  the  same  reason,  in  the  formula 

(±^±<^)X(it:0  =  (±^)X(dzO+(±^)X(±0> 
X  (±:^)  may  be  replaced  by  /  (±:^),   giving  the  corre- 
sponding distributive  law  with  the  sign  of  reciprocation : 

(d=  ^  ±  ^)  /  ( =b  O  =-  (±  «)  /  (=t  0  +  (±  ^0  /  ( ±  0. 


38  LAWS    OF   ALGEBRAIC    OPERATION. 

But  while  the  sign  X  is  distributive  over  the  successive 
terms  of  a  sum  (Art.  20),  that  is: 

X  (±^±1^  ±:...)=X  (±^)+  X  (lii^)-f  X.  .  ., 

the  sign  /  is  not,  as  may  be  readily  seen  by  constructing 
the  product  c  X  /  (^  +  ^),  and  the  sum  of  products  c  y' a 
-f-^  X  lb,  and  comparing  the  results,  which  will  be  found 
to  differ. 

22.     Agenda:     Theorems  in  Proportion,  Arithmetical 
Multiplication  and  Division. 


(I).  If 

A  .B  ::C  :D 

y.E  :F::.  .  . 

, 

prove  that 

A:B: 

\A  4-  C+^-f 

.  ..  :B  -\-D-\-F-^  ... 

(Adde7ido.) 

(2).     If 

A  .B  : 

.C  :D, 

prove  that 

A  +^  :B  : 

.C^  D  .D 

(Co77ipoiie?ido.) 

and  that 

Ar^B  :B  :: 

Cr^  D  :D. 

(  Dividendo. ) 

(3)-     If 

A  :B  :; 

c.nr 

prove  that 

A  ^  B  ■  A  ^  B  ■:  C  -^  D  ■  C  ^.  D. 

(4).      If       A.B  y.P  :Q,     B  :  C  ::  Q  :  R, 

C  :D  ::B  .S,  and  Z^  :  E  ::  S  :  T, 
prove  that 

A  :E  '.:P  :T.  (Ex  cequali.) 

(5).      If     A  '.B -..Q-.R^xiAB  .C  v.P  -.Q, 
prove  that 

A  -C  v.P  :R. 

(6).      Show  that  corresponding  to 

^X(X^-fX-^)=X(X^X^+X^X^) 


LAWS    OF   ALGEBRAIC    OPERATION.  39 

there  is  the  analogous  formula 

cK!a  +  lb)^j{lcla-\-  Iclb). 

(7).  Construct  the  product  3X2  and  show  that  the 
result  is  6. 

(8).  Construct  the  quotient  i  2  /  3  and  show  that  the 
result  is  4. 

(9).      Construct  5  /  3  and  2X5/3- 

vii.     Exponentials  and  Logarithms. 

23.  Definitions.'*'  Suppose  F,  Q  to  be  two  points 
moving  in  a  straight  line,  the  former  with  a  velocity,  more 
strictly  a  speed, "^"^^  proportional  to  its  distance  from  a  fixed 
origin  O,  the  latter  with  a  constant  speed.     Let  x  denote 

O  J  P P P' 

Q  Q  Q 

Fig.  20. 

the  variable  distance  of  P,  y  that  of  Q  from  the  origin,  and 
let  A.  be  the  speed  of  P  when  x=  (9/=  i,  /u,  the  constant 
speed  of  Q.  As  it  arrives  at  the  positions  /,  P\  P''  suc- 
cessively, P  is  moving  at  the  rates : 

A,     a:' X,     :r'' A,  respectively, 
the  corresponding  values  of  x  and  y  are : 

x--=OJ=^  I,  x'=  I  -\~JP\  x^'^i+JP'', 
y  =  o,         y'=OQ\         y''=OQ'' 


*  Napier's  definition  of  a  logarithm.  Napier  :  Mirifici  logarithmorum  canonis 
descriptio  (Lend.  1620),  Defs.  1-6,  pp.  1-3,  and  the  Construction  of  the  Wonderful 
Canon  of  Logarithms  (Macdonald's  translation,  1889),  Def.  26,  p.  19.  Also  Mac- 
Laurin:  Treatise  of  Fluxions,  vol.  i,  chap,  vi,  p.  15S,  and  Montucla:  Histoire 
des  Mathematiques,  t.  ii,  pp.  16-17,  97- 

**  The  speed  of  a  moving  point  is  the  amount  of  its  rate  of  change  of  position 
regardless  of  direction.  Velocity  takes  account  of  change  of  direction  as  well 
as  amount  of  motion.  See  Macgregor  :  Element aiy  Treatise  on  Kinematics  and 
Dynamics,  pp.  22,  23  and  55. 


40  LAWS    OF    ALGEBRAIC    OPERATION. 

and  Q  is  supposed  to  pass  the  origin  at  the  instant  when  P 
passes  y,  at  which  point  x  =  0/=  i.  The  speed  of  P 
relatively  to  that  of  Q,  or  vice  versa,  is  obviously  known 
as  soon  as  the  ratio  of  /x  to  A  is  given.  By  means  of  this 
construction  the  terms  modulus,  base,  exponential,  and 
logarithm  are  defined  as  follows : 

(i).  /LL  /A,  a  given  value  of  which,  say  m,  determines  a 
system  of  corresponding  distances  .x',  x^\  .  .  .  and^',  jv",  .  .  . 
is  called  the  modulus  of  the  system. 

(ii).  The  modulus  having  been  assigned,  the  value  of 
X,  corresponding  to  _y=  0J=^  i,  is  determined  as  a  fixed 
magnitude,  and  is  called  the  base  of  the  system.  Let  it  be 
denoted  by  b. 

(iii).  x^  x\  x^\  .  .  .  are  called  \h^  exponeiitials  oi y\  y' , 
y'\  .  .  .  respectively,  with  reference  either  to  the  modulus  m 
or  the  base  b,  and  the  relation  between  x  and  y  is  written 

X  ^  exp,;^  y ,     or  x  ^  b  y .  *^ 

(iv).  y,  y\y'\  •  •  •  are  called  the  logarithms  of  x,  x\ 
x^\  .  .  .  respectively,  with  reference  to  the  modulus  in,  or 
the  base  b,  and  the  symbolic  statement  of  this  definition  is 
either 

y  ^  log;;,  X,     or  y^  ^\og  x. '-!' 

The  convention  that  F  shall  be  at  /  when  Q  \<,^i  O  intro- 
duces the  convenient  relations 

log,;;  I  =  o,     and  exp,;,  o  =  i ,  or  b^  =  j, 

and  because  (by  definition)  y  =  i  when  x  =  b, 

log,;;  b  =■  I,      and  exp;;,  ^  =b,  or  b'  =^ b. 


*  ^\og  X  is  the  German  notation.  English  and  American  usage  has  hitherto 
favored  writing  the  base  as  a  subscript  to  log  thus,  log^  ;  but  Mr.  Cathcart  in  his 
translation  of  Harnack's  Differential  u.  Integral  Rechming  has  retained  the 
German  form,  which  is  here  adopted  as  preferable 

**See  Appendix,  page  139. 


LAWS    OF    ALGEBRAIC    OPERATION.  41 

Exponentials  and  logarithms  are  said  to  be  mvcf^se  to 
each  other. 

(v).  The  logarithms  whose  modulus  is  unity  are  called 
natural  logarithms,^  and  the  corresponding  base  is  called 
natural  base,  the  special  symbols  for  which  are  In  and  e 
respectively;  thus 

y  =  ^ogi  X  =  In  .T,     and  x  =  ey  =  exp  y 

represent  the  logarithm  and  its  inverse,  the  exponential,  in 
the  natural  system. 

24.  Relations  between  Base  and  Modulus.  Let 
the  speed  of  P  remain  equal  to  \x  as  in  Art.  23  while 
the  speed  of  Q  is  changed  from  /u,  to  k^k.  The  modulus, 
^IX  =  f/i,  will  then  be  changed  to  k  jx  j  \  =  k  7/1,  and  the 
distance  of  Q  from  the  origin,  corresponding  to  the  distance 
X,  will  become  ky.      Hence 

ky  =  Xogkvi  ^^  =  k  log;;,  X  ; 
that  is :      To  multiply  the  7nodulus  of  a  logarithm  by  any 
real  quantity  has  the  effect  of  multiplying  the  logarith7n 
itself  by  the  same  quantity. 
In  particular  we  may  write 

log;;/  -'^'  =  ^^^  In  X. 
Corresponding  to  this  relation  between  logarithms  in  the 
systems  whose  moduli  are  in  and  km^  the  inverse,  or  expo- 
nential relation  is 

^^Vkm  ky  =  exp;;,  y  =  exp„,  (|  ky). 


*  These  are  also  sometimes  called  Napierian  logarithms,  but  it  is  well  known 
that  the  numbers  of  Napier's  original  tables  are  not  natural  logarithms.  The 
relation  between  them  is  expressed  by  the  formula 

Napierian  log  of  ;f  =  lo"  In  (lo^  /  .r). 
Napier's  system  was  not  defined  with  reference  to  a  base  or  modulus.     See  the 
article  on  JS-apier  by  J.  W.  L.  Glaisher  in  the  Encyclopcsdia  Britannica,  ninth 
edition. 


42  LAWS  OF  ALGEBRAIC  OPERATION. 

Let  c  be  the  base  in  the  system  whose  modulus  is  km;  then 
since  7=1  when  x  =^b,  and 

ky  =  \o%km  ^^  =  log^„,  ^  =  I ,  when  x  =  c, 
,'.     b  =  exp^;«  k  =  c^. 
and 

Hence :     If  the  modulus   be  changed  froDi  m  to  kvi,  the 
corresponding  base  is  chajiged  from  b  to  b^^^. 

Again,  if  in  the  equation  log^^  x  =  in  hi  x,  b  be  substi- 
tuted for  X,  the  value  of  m  is  obtained  in  the  form 

m=  I  I  In  b. 

Hence  In  terms  of  the  base  of  a  system  the  logarithm  is 

logw  ^  =  h^  ^  I  hi  b, 

and  if  to  x  the  value  e  be  given,  m  is  obtained  in  the  form 

m  =  log„,  e  ; 

whence  passing  to  the  corresponding  inverse  relation : 

exp„2  7n  =  b'"  ^=  e  ; 

that  is :      The  exponential  of  any  qiiaiitity  with  respect  to 
itself  as  a  modiilns  is  equal  to  natural  base. 

25.  The  Law  of  Involution.  By  virtue  of  the 
fundamental  principle  of  the  last  article — that  to  multiply 
the  modulus  multiplies  the  logarithm  by  the  same  amount — 
we  have  in  general 

exp;.;;,  z  =  exp„,  z  /  k 
and 

exp,„/^sr  =  exp,«/^2:. 
Hence 

exp,„  hk=^  exp;„  7,  k  =  exp„,  ^  //  =  exp;„/M  i; 

*  For  an  amplification  of  this  proof  see  Appendix,  page  139. 


LAWS    OF   ALGEBRAIC    OPERATION.  43 

or  ill  other  terms,  since  changing  ;//  into  vi  /  //,  or  ni  /  k^ 
changes  b  into  M,  or  h^^, 

which  expresses  the  law  of  involution.  Obviously  //,  or  k, 
or  both,  may  be  replaced  by  their  recii)rocals  in  this  formula 
and  the  law,  more  completely  stated,  is 

Evolution.  If  /^  be  an  integer,  the  process  indicated  by 
b^'^  is  called  evolution;  and  when  k^=2^  it  is  usually 
expressed  by  the  notation  -j/  b. 

26.  The  Law  of  Metathesis.  Let  z-r=b^^\  then 
by  the  law  of  involution 

whence,  passing  to  the  corresponding  inverse  relations, 

/ik  =  log„^  zk 

and 

h  --=  log;;,  z  ; 

.    •.  k    log;;,     Z   =    log;;,     5^^' , 

which  expresses  the  law  of  interchange  of  exponents  with 
coefficients  in  logarithms, — the  law  of  metathesis. 

27.  The  Law  of  Indices.*  The  law  of  indices,  or 
of  addition  of  exponents,  follows  very  simply  from  the  defi- 
nition of  the  exponential,  thus:  In  the  construction  of 
Art.  23,  and  by  virtue  of  Definition  (iii )  of  that  Article 

O J P F P" 

Q  Q  Q 

Fig.  20. 

exp;;,  /=  I  +  JP\      exp;;,  f'=  I  +  JP^' 

.   \  eXP;;,    j/;/    eXp;;,   j'    =    OP''/    OP' 

=  1  4~P'P''   OP\ 


For  an  alternative  proof  see  Appendix,  pages  139-141. 


44  LAWS    OF    ALGEBRAIC    OPERATION. 

But  the  magnitude  i  —  F'F'^  OP^  represents  the  distance 
that  P  would  traverse,  starting  at  unit's  distance  to  the  left 
of  P\  on  the  supposition  that  its  speed  at/'' is  \x' j x'=^\^ 
and  the  corresponding  distance  passed  over  by  Q  isy'^—y-, 
hence,  by  definition  of  the  exponential, 

I  +  P'P^r  0P'=  exp;,  (y''-y), 

and  therefore 

exp,„  y'\'  exprn  y=exp„,  (y'—y'), 

which  is  the  law  of  indices  for  a  quotient.      In  particular,  if 

y^=o 

exp;;,  o     exp;;,  ^'=  exp„,  (o  —  ^') 
or 

I  /  exp;;,  y=  exp;;,  ("J'')  ;  (Art.  23.) 

and  therefore,  writing  — y'r^y^  the  foregoing  equation, 
exp;„  7"/exp„,7'=  exp;;,  {^y'  —  y'),  becomes 

exp;;,  y ' '  X  exp„,  y  =  exp^;,  (  7  "  ^  7) , 

which  is  the  law  of  indices  for  a  direct  product.  The  com- 
plete statement  of  this  law  is  therefore  embodied  in  the 
formula 

exp;;,  y  y/  exp„,  y'  ^  exp,,,  (7''  ±y'), 

or  its  equivalent 

byy/by=by'-y. 

28.     The  Addition  Theorem.     Operating  upon  both 
sides  of  the  equation  last  written  with  log;;^  we  have 

log,.  (^-^'">^^-^')=y'=^y; 

that  is,  replacing  1^',  by  by  x'',  x'  2indy'\  y'  by  log;;,  x'\ 

l0g;;;:v', 

log;;,    {X''    Yy    X'  )=.    log,;,   x' '  ±    lOg,;,    X\ 

which  is  the  addition  theorem  for  logarithms. 


LAWS    OF   ALGEBRAIC    OPERATION.  45 

29.  Infinite  Values  of  a   Logarithm.     If,   in  the 

construction  of  Art.  23,  X  and  /a  become  larger  than  any 
previously  assigned  arbitrarily  large  value,  while  their  ratio 
m  (that  is,  the  modulus)  remains  unchanged,  P  and  Q  are 
transported  instantly  to  an  indefinitely  great  distance,  and 
OP,  OQ  become  simultaneously  larger  than  any  assign- 
able magnitude.  It  is  customary  to  express  this  fact  in 
brief  by  writing 

b'^^OO,  log;;,     00=    CO     ; 

though  to  suppose  these  values  actually  attained  would 
require  both  A  and  ^u,  to  become  actually  infinite.  This  sup- 
position will  be  justifiable  whenever  we  find  it  legitimate, 
under  the  given  conditions,  to  assign  to  the  indeterminate 
form  /a  ■  A.  :=  00  /  00  a  determinate  value  m.-^^ 

In  like  manner,  since  from  d^  =  co  we  may  infer 
^— =^  =  I  /  CO  =  o,   the  equations 

^  — °^  =  o,     log;;,  o  =  —  00 

are  employed  as  conventional  renderings  of  the  fact,  that 
when  P  and  Q  are  moving  to  the  left,  P  passing  from  / 
towards  O  and  Q  negatively  away  from  O,  x,  in  the  equa- 
tions X  =  l?^ ,  y  =  log;;,  X,  remains  positive  and  approaches 
o,  while  y  is  negative  and  approaches  —  co . 

30.  Indeterminate  Exponential  Forms.  When 
^  X  logw  tf  becomes  either  ±0  X  ^,  or  ±  00  X  o,  it  is 
indeterminate  (Art.  11).  Now  log;;,  ?/  is  o  if  ?/=i,  is 
-j-  00  if  u=  -^  00,  and  is  —  co  if  u  =  o  (Art.  29).  Hence 
V  X  log;;,  z/  will  assume  an  indeterminate  form  under  the 
following  conditions : 

*  This  form  of  statement  must  be  regarded  as  conventional.  Strictly  speak- 
ing we  cannot  assign  a  value  to  an  indeterminate  form.  When  the  quotient  x  j  y, 
in  approaching  the  indeterminate  form,  remains  equal  to,  or  tends  to  assume,  a 
definitive  value,  we  substitute  this  value  for  the  quotient  and  call  it  a  limit.  In 
conventional  language  the  indeterminate  form  is  then  said  to  be  evaluated. 


46  LAWS    OF   ALGEBRAIC    OPERATION. 

When  z'  =  00  and  71=^  i, 
or  e^  =  o   and  ?/  =  -j-  c)o> 
or  ?^  =  o    and  z/ --=  o. 

But  if  z>  X  log,«  ?/,  or  \og„i  u^  is  indeterminate,  so  is  ?/^',  and 
therefore  the  forms  i  °°,  00'',  0°  are  indeterminate. 

Whenever  one  of  these  forms  presents  itself,  we  write 
y  =  u-<J  and,  operating  with  In,  examine  the  form 

In  j;  =  z'  X  hi  u. 

If  then  In  7  can  be  determined,  y  can  be  found  through  the 
equation 

y  z=  ^In  y^ 

viiL     Synopsis  of   Laws  of  Algebraic  Operation.* 

31.     Law  of  Signs: 

(i).     The  concurrence  of  hke  signs  gives  the  direct  sign, 
-f  or  X.     Thus  : 

+  +  =  +,    —  =  -f,    xx  =  x,    //=x. 

(ii).     The  concurrence  of  unHke  signs  gives  the  inverse 
sign,  —  or  /.     Thus : 

(iii).      The  concurrence  of  two  or  more  positive,  or  an 
even  number  of  negative  factors,  in  a  product  or  quotient, 
gives  a  positive  result.     Thus  : 
(+  a)y/(^-\-b)  =  -^ay9b,       (^-a)y/{-b)  =  -ay/  b. 

(iv).     The  concurrence  of  an  odd  number  of  negative 
factors,  in  a  product  or  quotient,  gives  a  negative  result. 
Thus: 
i^-  ci)^  i—  b')  =-  —  aY/  b ,        {—  a)  y^  (i-\-  b)  =  —  ay</  b. 


Cf.  Chrystal:     Algebra,  vol.  I,  pp.  20-22. 


LAWS    OF    ALGEBRAIC    OPERATION.  47 

32.     Law  of  Association: 


For  addition  and  subtrac- 
tion : 

zt  (zt  a  ±1  d) 


For  multiplication  and  di- 
vision : 

=  y/(y/a)y/{y/b). 


33.     Law  of  Commutation  : 


For  addition  and  subtrac- 
tion: 


For  multiplication  and  di- 
vision : 

Y/aY/b^y/by/a. 


34.  Law  of  Distribution: 

( i ) .      For  multiplication : 

=  +  (±^)  X  (±^)-f  (it^)  X  (±^) 
4-  (dz  ^)  X  (±  c)  +  (±  b)  X  (±  d-). 
(ii).      For  division: 

(±  ^  ±  ^)  /  (±  0  =  -f-  (±:  «)  /  (±:  0  4-  (±  ^)  /  (±  0- 
A  divisor  consisting  of  two  or  more  terms   cannot  be 
distributed  over  the  dividend. 

35.  Laws  of  Exponents: 

(i).      Involution  :      («^/"0  ^/«  -=  («^«)  ^'^ 
(ii).     Index  law  :     Yy  a'"  Yy  a"  =  a-'^^'K 
Also,  as  a  consequence  of  these  two : 

(iii).    Corollary :     {yy  aYy  by  =  Y/  «"'  Y/  b»K 

36.  Laws  of  Logarithmic  Operation: 

(i).      Metathesis: 

n  \ogjn  X  =  m  log  ;,-v  =  log;„  x"  =  In  x""\ 
(ii).    Addition  theorem : 

log„,  (-r  >^  jO  =  log,;,  .r  ±  log,„  J/. 


48 


LAWS  OF  ALGEBRAIC  OPERATION. 


37.     Properties  of  o,   i,  and  c»: 

0=  -\-  a  —  a, 
ztd  ^0=  ±^  —  o, 
—  0=  —  o, 

±00  zlz  ^  =  =b  00, 


I  =  >■.;  a  I  a, 
Y/by^  i=y/b!i, 
X  I  =    I, 
o>^  ^  =  0, 

b    CO  =  X  o. 


Zero  may  be  regarded  as  the  origin  of  additions,  unity 
as  the  origin  of  multiplications. 

38.  Agenda:  Involution  and  Logarithmic  Operation 
in  Arithmetic. 

(i).  Show  that,  if  n  be  an  integer,  the  index  law 
(Art.   35)  leads  to  the  result: 

«« z=:  ^  X  <^  X  ^  X  •  .  .  .  to  7z  factors, 

and  that  therefore  3^  =  9,  2^  =  8,  5^  =  1 25,  etc. 

(2).  vShow,  by  the  law  of  involution  and  the  index  law 
(Art.  35),  that  8^/^=2,  8i^''*=3,  etc. 

(3).  Show,  by  the  law  of  metathesis  (Art.  36),  that 
4og  32  =  5,  nog  729  =  6,  etc. 

(4).  Show,  by  the  law  of  metathesis  and  the  addition 
theorem  (Art.  36),  that 

^log  8  +  "^log  2  =  2,     "^log  2  =  ^, 
nog(i/8)-f  nog(i  /27)  =  -3. 

(5).  Find  the  logarithms:  of  16  to  base  2'-,  of  125 
to  base  5  X  5'  %  of  128  to  modulus  i  /  In  8,  of  i  /  81  to 
modulus  I  /  In  27. 


CHAPTER    II. 

GONIOMETRIC  AND   HYPERBOLIC   RATIOS. 


IX.       GONIOMETRIC    RATIOS. 

39.  Definition  of  Arc-Ratio.  In  the  accompanying 
figures  JV'JVis  a  straight  line  fixed  in  position  and  direction, 
OP  is  supposed  to  have  reached  its  position  by  turning 
about  the  fixed  point  O  in  the  positive  sense  of  rotation 
from  the  initial  position  OJV.     Any  point 


O      C  L   A    N 


JFtg-.  21. 


on  OP  at  a  constant  distance  from  O  describes  an  arc  A  VQ, 
a  Hnear  magnitude.  Let  the  ratio  of  this  arc  to  the' radius 
OQ,  both  taken  positively,  be  denoted  by  6,  that  is, 

^^  (length  of  arc  AVQ)  I  (line-segment  OQ). 

The  amount  of  turning  of  OQ,  that  is,  the  angle  AOQ, 
fixes  the  value  of  this  ratio ;  and  since  the  arcs  of  concentric 
circles  intercepted  by  common  radii  are  proportional  to  those 
radii  (Prop.  18),  the  ratio  may  be  replaced  by  an  arc  CD 
provided  only  OC  be  taken  equal  to  the  linear  unit.     In 


50  GOXIOMETRIC    AND    HYPERBOLIC    RATIOS. 

the  geometrical  figures  a  description  of  the  angle  will  be 
sufficient  to  identify  the  ratio  itself.  This  magnitude  6  will 
be  called  the  arc-ratio  of  the  angle  AOQ.  The  letter  tt 
stands  for  ratio  of  a  semi-circumference  to  its  radius,  that 
is,  the  arc-ratio  of  i8o°. 

Lines  drawn  parallel  or  perpendicular  to  A''iV,  shall  be 
regarded  as  positive  when  laid  off  from  O  to  the  right  or 
upwards,  negative  when  extending  to  the  left  or  downwards. 
OP  drawn  outwards  from  O  is  to  be  considered  positive  in 
all  cases. 

40.     Definitions  of  the  Goniometric  Ratios.     LQ 

in  the  above  figures  being  drawn  perpendicular  to  N' N, 
upwards  or  downwards  according  as  Q  is  above  or  below 
N^ N  and  correspondingly  positive  or  negative,  the  gonio- 
metric ratios,  called  sine,  cosine,  tangent,  cotangent,  secant, 
cosecant,  are  defined  as  functions  of  the  arc-ratio  B  by  the 
following  identities : 

sin  6  =  LQI  OQ,  cos  0=OL  OQ, 
tSLnO  =  LQ  /  OL,  cot  0=OL  I  L Q, 
sec  0=  OQ    OL,     esc  6=  OQ    LQ. 

It  must  be  borne  in  mind  that  6  is  here  not  an  angle 
expressed  in  degrees,  but  a  ratio,  which  can  therefore  be 
represented  by  a  linear  magnitude.  In  elementary  trigo- 
nometry sin  6  usually  means  ''sine  of  angle  AOQ  in 
degrees";  here  it  may  be  read  "sine  of  magnitude  ^," 
where  0  =  arcAVQ/  radius  OAr^  If  v  be  the  number  of 
degrees  in  the  angle  AOQ,  the  relation  between  v  and  6  is 

180^.  (Prop.  17.) 


TTZ' 


See  Lock's  Elementary  Trig onojne try,  p. 


GONIOMETRIC   AND    HYPERBOLIC    RATIOS. 


51 


41.    Agenda. 

Properties  of  Goniometric 

e  following : 

(0- 

sin  0^=0,      cos  0=1. 

(2)- 

sin  f  =  I,     cosf  =  0. 

(3)- 

sin^^  +  cos^^="i. 

(4)- 

sec^^— tan^^=i. 

(5). 

csc^O  —  COt^^=I. 

If  n  be  an  integer,  prove : 

(6).     sin  (0  ±  [2  n  +  i]  tt)  =  =b  cos  0. 

(7).     cos  ((9ih  [2w +  i]7r)  =  q=sin  ^. 

(8).     sin  (0±  [2n  -|-|]7r)  =  q=  cos  ^. 

(9).  cos  (^  ±  [2  ;2  +  I]  tt)  =  dz  sin  0. 
(10).  sin,  or  cos  of  (^  zb  2  n  tt)  :=  sin,  or  cos  of  0. 
(11).     sin,  or  cos  of  (0  zb  [2  n  -f  i]  tt)  =  —  (sin,  or  cos  of  0). 

42.    Line-Representatives  of  Goniometric  Ratios. 

If  in  the  foregoing  definitions  the  denominators  OL,  LQ 
be  replaced  by  the  radius  OQ^  the  numerators  of  the  six 
goniometric  ratios  will  be  six  straight  lines  drawn,  either 
from  the  centre,  or  from  Q,  or  from  one  of  the  fixed  points 
A^  B  on  the  circumference  a  quadrant's  distance  apart. 


S^P 


Fig.  22. 

If  a  be  the  radius  of  the  circle,  they  may  be  Indicated 
as  follows : 


52 


GONIOMETRIC    AND    HYPERBOLIC    RATIOS. 


«  sin  6=^LQ,  perpendicular  distance  of  Q  from  A' A. 
a  cos  B=  OL,  distance  from  centre  to  foot  of  LQ. 
a  tan  6  =  AT,   distance  along  a  tangent  from  A  to  OF. 
a  cot  0  =  BS,  distance  along  a  tangent  from  B  to  OF, 
a  sec  0  =  OM,  intercept  of  tangent  at  Q  upon  OA. 
a  CSC  0=  ON,  intercept  of  tangent  at  Q  upon  OB, 

These  constructions  are  evidently  only  variations  in  the 
statement  of  the  definitions  of  the  goniometric  ratios. 
When  ti=  I,  the  six  ratios  have  as  their  geometric  rep- 
resentatives these  lines  themselves. 

Formerly  they  were  defined  as  such  for  all  values  of  the 
radius  and  were  therefore  not  ratios,  but  straight  lines 
dependent  for  their  lengths  upon  the  arc  AQ,  that  is  upon 
both  the  angle  AOQ  and  the  radius  of  the  circle.  The 
older  form  of  definition  is  now  rare.* 


43.     To  Prove  Limit  [(sin  ^) /^]  =  i,  when  B^^o. 
Let  ^^the  arc-ratio  of  the  angle  FOQ  in  Fig.  23,  draw 

FQF\  an  arc  with  radius  OF, 
draw  PT'and  F^l''  tangents  to 
the  arc  at  F  and  F' ,  join  F,  F\ 
and  O,  T.  Then  assuming 
that  an  arc  is  greater  than  the 
subtending  chord  and  less  than 
the  enveloping  tangents  at  its 
Fig.  23.  extremities,  we  have 


2  SF<2QF<^2FT, 


'''SP<SP<ZSP^ 


that 


^  sin  0  . 
I  >  -^  >  cos 


*  See  Todhunter's  Plane  Trigonometry,  p.  49,  and  the  reference  there  given: 
Peacock's  Algebra,  Vol.  II,  p.  157.  See  also  Buckingham's  Differential  and 
Integral  Calculus,  3d  ed.,  p.  139,  where  the  older  definitions  are  still  retained. 


GONIOMETRIC    AND    HYPERBOLIC    RATIOS.  53 

When  ^  =  o,  cos  6=1]  therefore  by  making  6  smaller  than 
any  previously  assigned  arbitrarily  small  magnitude,  (sin  0)  jO 
is  made  to  differ  from  unity  by  a  like  arbitrarily  small  magni- 
tude. Under  these  conditions  (sin  0)  /  0  is  said  to  have  i  as 
its  limit,  and  the  fact  is  expressed  by  the  formula 

limit  ( sin  6 


in  which  =  stands  for  'approaches.' 

44.  Area  of  Circular  Sector.  Let  the  sector  OAQ 
be  divided  into  n  equal  smaller  sectors  by  radii  to  the  points 
I*^,  /*2,  i^3,  etc.,  which  set  off  the  arc  AQ  into  the  same 
number  of  equal  parts  AF^,  P^P^, 
Pr,P^,  etc.,  and  draw  P^Al  perpen- 
dicular to  OA.  The  area  of  each  of 
the  triangles  OAP,,  OP^P.^,  OP,P^, 
.  ,  .\s\a- MP, , 

orif  (arc^(2)/C>^  =  ^,   it  is 
\a'asm{AP,lOA^  =  \a'-;\n{Bln), 
and   hence   the   area   of  the   entire 
^'^-  '^-  polygon  OAP,  P, .  .  .  Q  \s 

na'     .    0       a'O    s\n  (0 1  n) 

•  sm  -  =  —  •  — n-i . 

2  n         2  d  I  n 

Now  when  the  number  of  points  of  division  P,,P^,P^,  .  .  . 
is  indefinitely  increased,  the  polygon  OAP^P^  -  •  •  Q  ap- 
proaches coincidence  with  the  circular  sector  OAP^P^  •  -  •  Q-. 
that  is, 

—  •  — -^r^ — ^  r=  area  of  sector  OAP^P^  -  -  -  Q,  when  «  =  00 ; 

but  at  the  same  time 

sin  {OJji) 
0  /  nzrzo  and  ""  ^  .  ^^       =zi, 


54  GONIOMETRIC    AND    HYPERBOLIC    RATIOS. 

and  therefore  also 


e  1 71 


,  when  71  =  o:>. 

2 


Thus  —  •  — rrr —  can  be  made  to  differ,  both  from  — 
2  6  j  n  '  2 

and  from  the  area  of  the  circular  sector,  by  quantities  that 
are  less  than  any  previously  assigned  arbitrarily  small 
magnitude.  Under  these  circumstances  it  is  assumed  as 
axiomatic  that  the  two  limits  which  the  varying  quantity 
approaches  cannot  differ,  and  that  therefore 


area  of  sector  OAP^F^ 


^  2 


The  limits  in  fact  could  not  be  different  unless  the  area  of 
the  sector  were  susceptible  of  two  distinct  values,  which  is 
manifestly  impossible. 


45.     Agenda.     The  Addition  Theorem  for  Goniometric 
Ratios. 

From  the  foregoing  definitions  of  the  goniometric  ratios 
prove  for  all  real  arc-ratios  the  following  formulae : 


(T). 

sin  (a  ±  /8)  =:  sin  a  cos  /8  ±  cos  a  sin  /3. 

(2). 

cos  (a  ±  /3)  =  cos  a  cos  ji  +  sin  a  sin  /3. 

(3). 

tan  a  ±  tan  6 
t^"(«±«-i+tanatan^- 

(4)- 

sin  2  a  =  2  sin  a  cos  a. 

(5). 

cos  2  a  =  cos""  a  —  sin^  a. 

(6). 

I  +  cos  2  a  =  2  COS^  a. 

(7). 

I  —  COS  2  a  =  2  sin^  a. 

GONIOMETRIC    AND    HYPERBOLIC    RATIOS. 


55 


X.     Hyperbolic  Ratios. 

46.     Definitions  of  the    Hyperbolic    Ratios.     An 

important  class  of  exponentials,  which  because  of  their  rela- 
tion to  the  equilateral  hyperbola  are  called  the  hyperbolic 
sine,  cosine,  tangent,  cotangent,  secant  and  cosecant,  and 
are  symbolized  by  the  abbreviations  sinh,  cosh,  tanh,  coth, 
sech,  csch,  are  defined  by  the  following  identities : 

sinh  u^^  (e"  —  ^~"), 

cosh  2/^^  (^"  -f  e-"), 

tanh  «  =  (^"  —  e~")  /  (e"  -f  e-"), 

coth  u^ie^'  +  ^-")  /  ((?"  —  ^-"), 

sech  u^2  I  {e^  -{-  ^~"), 

csch  u^2  I  (^"  —  ^~"). 


47.     Agenda.     Properties  of  Hyperbol 

the  following : 

(I) 

sinh  0  =  0,      cosh  0=1. 

(2) 

cosh^  u  —  sinh^  ti  =  1. 

(3) 

sech"  w  +  tanh"  z^  =  i. 

(4) 

coth"  u  —  csch"  u=  1. 

(5) 

sinh  2  u  =  2  sinh  7i  cosh  u. 

(6) 

cosh  2  u  =  cosh"  u  -\-  sinh"  u 

(7) 

cosh  2  u  -\-  1  =  2  cosh"  u. 

(8) 

cosh  2  u  —  I  =  2  sinh"  u. 

(9) 

sinh  ( —  7c)  =  —  sinh?/. 

(10) 

cosh(—  21)  =  cosh  2t. 

48.  Geometrical  Construction  for  Hyperbolic 
Ratios.  For  the  representation  of  the  hyperbolic  ratios 
the  equilateral  hyperbola  is  employed.  Its  equation  in 
Cartesian  co-ordinates  is 


y-=a^ 


56 


GOXIOMETRIC    AND    HYPERBOLIC    RATIOS. 


Let  OX,  6>  F  be  its  axes,  OJ  an  asymptote,  P  any  point  on 
the  curve,  x  and  y  its  co-ordinates,  ^(2^  the  quadrant  of 


a  circle  with  centre  at  the  origin  and  radius  a,  NQ  a 
tangent  to  the  circle  from  the  foot  of  the  ordinate  y,  PS 
a  tangent  to  the  hyperbola  parallel  to  the  chord  FA,  a,  /3 
the  co-ordinates  of  (2,  0  the  arc-ratio  of  the  angle  ACQ. 

It  is  obvious  from  its  definition  that  cosh  u  has  i  for  its 
smallest  and  -j-  oo  for  its  largest  value  corresponding  to 
21  =  o  and  CO  respectively,  and  if  the  variations  oi  x  j  a  be 
confined  to  the  right  hand  branch  of  the  hyperbola  its  range 
of  values  is  likewise  between  i  and  +  ^  ',  hence  we  may 
assume 

—  =  cosh  u, 


GONIOMETRIC    AND    HYPERBOLIC    RATIOS.  57 

and  by  virtue  of  the  relations  cosh^  it  —  sinh^  ic=  i,   and 

x'  /  a^  —  y  !  a-^=  I, 

y 

•^  =  sinh  ti. 
a 

Also,  since  x^  —  y~  z=  a"  and  x"  —  NQ"  =  a-,  therefore 

JVQ  =  NF, 
and  we  have 

X  V 

—  =  sec  6  =  cosh  7c,      -  =  tan  6  =.  sinh  u. 

and  the  co-ordinates  of  Q  being  a,  )8,  also 

AH      ^      y        '     r\  1  (X       a  f.  , 

=  -  =  ^  =  sni  u  =  tanh  21,      -  =  -  =  cos  u  =  sech  ?^, 

a         a       X  ax 

and  finally,  since  OBK  is  similar  to  6>A^i^  and  a-  =^y  '  LO^^ 

BK      X  ,  LO       a 

=  —  =  coth  u,     =     =  csch  2C. 

ay  'ay 

Hence  if  a  be  made  the  denominator  in  each  of  the 
hyperbolic  ratios,  their  numerators  will  be  six  straight  lines, 
drawn  from  O,  A,  B,  or  P,  which  may  be  indicated  thus: 

a  sinh  ii  =  NP,  perpendicular  distance  of  P  from  OX, 
a  cosh  u  =  ON,  distance  £'om  centre  to  foot  of  NP, 
a  tanh  ti  =  AH,  distance  along  a  tangent  from  A  to  OP, 
a  coth  21  =  BK,  distance  along  a  tangent  to  the  conjugate 

hyperbola  from  B  to  OP, 
a  sech  21  ^=  OM,  intercept  of  tangent  at  P  upon  OX, 
a  csch  2c  =  LO,  —  (intercept  of  tangent  at  P  upon  OY^. 

This   construction  gives   pertinence   to   the   name   ratio  as 


*  Obtained  bywriting  the  equation  of  the  tangent  iJ/Pand  finding  its  intercept 
on  Oy\  or  thus.  OL  1  OM=NP  I  MN,  that  is,  OL  =  WM .  y)  I  {x  —  OM) 
=  a  (sech  ii  sinh  u)  I  (cosh  ji  —  sech  11)  =  a  sinh  u  j  (cosh^  u  —  i)-=  a  csch  w. 


58  GONIOMETRIC   AND    HYPERBOLIC    RATIOS. 

applied    to   the  six    analogues   of  the   goniometric  ratios. 
Compare  these  with  the  constructions  of  Art.   42. 

49.  Agenda.  Properties  of  the  Equilateral  Hyperbola. 
Prove  the  following  propositions  concerning  the  equilateral 
hyperbola.      (Fig.  25  of  Art.  48.) 

( I ).  The  tangent  to  the  hyperbola  at  P  passes  through 
M,  the  foot  of  the  ordinate  to  Q. 

(2).  The  locus  of  /,  the  intersection  of  the  tangents 
NQ  and  MP,  is  A  J  the  common  tangent  to  the  hyperbola 
and  circle. 

(3).  The  line  OIV  bisects  the  angle  0  and  the  area 
OA  VP  and  intersects  the  hyperbola  at  its  point  of  tangency 
with  RS. 

(4).  A  straight  line  through  P  and  Q  passes  through 
the  left  vertex  of  the  hyperbola  and  is  parallel  to  OV. 

(5).     The  angle  APN=  one-half  the  angle  QON". 

50.  The  Gudermannian.  When  6  is  defined  as  a 
function  of  2c  by  the  relation  tan  6^sinh  u  (Art.  48)  it  is 
called  the  Guderman7iian  of  «*  and  is  written  gd  u.  Sin  6, 
cos  0  and  tan  B  are  then  regarded  as  functions  of  u  and  are 
written  sg  n,  eg  21  and  tg  it. 

51.  Agenda.  From  the  definitions  of  the  Guder- 
mannian functions  prove  the  formulae: 


*  By  Cayley,  Elliptic  Functions,  p.  56,  where  the  equation  of  definition  is 
u  =  In  tan  (1  tt  +  J  Q). 

Since  tan  (i  ir  +  i  ^)  -=  ^  +  ^'"  ^  and  sinh-»  tan  ^  =  In  ^  "^  ^'"  ^  (Art.  56).  the 
cos  Q  cos  Q  "^ 

equivalence  of  the  two  definitions  is  obvious.     The  name  is  given  in  honor  of 

Gudermann,  who  first  studied  these  functions. 


GONIOMETRIC    AND    HYPERBOLIC    RATIOS.  59 

(l).       Sg'7l-{-Cg'2C=l. 

y   s           ^      ,     N          so;  21  -{-  so;  V 
(2).     sg  (u -i- v) --=^ — ^ — -^—^ . 

^     ^  ^  ^  -^  I  -j-  Sg7t  .  Sg  V 

^     N  r        ,       N  Cg7i.CgV 

(  3  ) .       Cg(2l  +  V)  =  —^ ^ . 

v.oy         s  V      I      y        I  -\-  sgic .  sgv 

(4).        If  2  =  ]/  -  I ,         V  gd0  gd  7^1  -=  —  2^, 

or  more  briefly  (~-g"<^)    ^^  =  —  ^^-  (Prof.  Haskell.) 

52.  To  Prove  Limit  [(sinh  ?/)  /  ?^]  =  i ,  when  2i  =  o. 
In  the  construction  of  Art.  23  suppose  that,  during  the 
interval  of  time  /' —  f,  P  moves  over  the  distance  x'  —  x,  Q 
over  the  distance  7^' —  ?^.  Then  speed  being  expressed  as 
the  ratio  of  distance  passed  over  to  time-interval,  the  speed 
of  (2  is 

O  J  P  P 


Q 

Q 

Fig,  26. 

2i'  -  U 

and  the  average  speed  of  P  during  the  whole  interval  is 


X 


Let  \  x^  represent  the  true  speed  of  T'  at  a  given  instant 
within  the  interval  considered,  \x^  —  8,  A  .To  -f  S'  the  speeds 
at  its  beginning  and  end  respectively ;  then 

\x,—  l<.  -  ^/  _  \  <  A  jr„  -f  S'; 

and  if  the  interval  /'  —  /  be  made  to  decrease  in  such  a  way 
that  S  and  o'  simultaneously  approach  zero,  the  three  mem- 


6o 


GONIOMETRIC   AND    HYPERBOLIC    RATIOS. 


bers  of  this  inequality  approach  a  common  value,  their  limit; 
that  is,  , 


/' 

—  t  ~ 

:a.^o,    u 

llCll    L 

I. 

Hence  also 

d"'- 

-b-  _ 

hen/' 

=t\ 

?/  - 

—  7t 

or  in  the  language 

of  limits  (Art. 

43). 

limit    id"' 

7C'=76   li^ 

—  7C    S    " 

A 

In  the  important  case  when  ?/  =  —  71,  and  7t  r=  o,  since 
u=^Q  requires  that  ,r  =  i ,  the  expression  last  written  be- 
comes 


limit 

7lr=0 


f)U  _  ^-« 


and  in  particular  when 


limit    j^"  —  <?-"! limit    (sinh?^ 

ti  =  0    (         2  21         )  7lz=:0    ^       7C 


=  I. 


O.  E.  D.      (Cf  Art.  43.) 

53.  Area  of  a  Hyperbolic  Sector.  Let  the  perpen- 
dicular p  be  dropped  from  any  point  F,  of  the  equilateral 
hyperbola  x^—y'=^d',   upon  its  asymptote,   meeting  the 

latter  in  S,  and  let  OS  =^  s. 
The  following  properties  of 
this  hyperbola  are  well 
known  and  are  proved  in 
elementary  works  on  conic 
sections: 

sp  =--  a'  I  2, 

X  —y=p^^^  _ 

x={s-\-p)li/  2, 


y  =  is-p)  l\ 


/ 


GONIOMETRIC    AND    HYPERBOLIC    RATIOS.  6 1 

Assuming  these  equations  as  known,   lay  off  upon  the 
asymptote 

OS,,  OS,,  0S._,  .  .  .,  0S  =  a/y2',s,,s._,  ,  .  . ,  s, 

such  that  these  lengths  are  in  geometrical  progression,  and 
let  the  corresponding  perpendiculars  upon  the  asymptote  be 

AS,,  F,S,,  F._S,,  .  ..,FS=p„p„p._,  ...,/. 

Then  if  p  be  the  common   ratio  of  the  successive  terms 
s^,  s^,  s^,  .  .  .  s,  we  have 

s,^=a  '■'  y'  2,     po  =  ci  !  \/  2, 
s,  =  ps,,  p,  =  a'  /  (2  ps,), 

s.  =  p"s,,  p^_=za-  I  {2  p's,). 


s  =  p''s,,         p  =  a- 1  (2  /3"  s,). 

If  now  (^,  o),  {x„  y,),  (x„  y,),  .  .  .  ,  (x,  y)  be  the 
co-ordinates  oi  A,  F,,  F^,  .  .  .  ,  F,  the  area  of  the  triangle 
OF^F^  is  \  (^x.^y^  —  x^y^),  and  by  virtue  of  the  relations 

X  =  {s-\-p)l^/2,    y  =  (^s-p)lV2, 

we  have 

but 

J3,  s,  =  ps„  ps,  and/,,  p^=pj p,  pj p; 

. • .    I  {^^,y.  -  ^.y,)  =  i  (s,p,  —  s^p,) 
=  i  (^.yr  -  x.y.) 

=  area  of  triangle  OF^F^. 

Thus  the  triangles  OAF,,  O  F,P,,  OP.^P^,  etc.,  are  equal 
to  one  another  in  area  and  the  area  of  the  whole  polygon 
0AP,P._.  ...  /^  is 

2  ^'^^^  -  ^°^^)  =  2  V^  7-2  -^'  V~2/ 


62  GONIOMETRIC   AND    HYPERBOLIC    RATIOS. 

But 


s  -1/2      X   ,    y 
^  a  a    '    a 

hence,  assuming  x  /  a  =  cosh  7c   and  y  /  a  =  sinh  7i, 


.  •.     Area  of  OAP, .  .  P  =  ~   (e"  '  "  —  e-"  '  ^) 
a"  u  ^  sinh  (?^  /  71) 

When  the  number  of  points  of  division  S^,  S^,  S\,  etc. ,  is 
indefinitely  increased  the  polygon  OAP^  P^.  .  P  approaches 
coincidence  with  the  hyperbolic  sector  OAP^  P^ .  .  P,  that  is, 

a-  u    sinh  (?/  /  71)  .^  ^  r^        t^     , 

——  '  -: z=z  area  of  sector  OriP,  .  .  P,  when  ;;,  -^  00  • 

2  U      71  I  >  , 

but  at  the  same  time 


,  sinh  (u  /  71)   . 
=  o  and  ^-7— —  = 

and  therefore 


tc  I  7i^^o  ana  '-j— — '-  =1, 

U  I  71  ' 


a?  2C    sinh  (u  /  71)       a"  it       , 

—  • -, =r  — ,  wnen  7i-=zoo. 

2  U  I  71  2    ' 

Hence  by  the  reasoning  of  Art.  44, 

area  of  sector  OAP,  P^ .  .  P= . 

2 

54.  Agenda.  The  Addition  Theorem  for  Hyperbolic 
Ratios.  From  the  foregoing  definitions  of  the  hyperbolic 
ratios  deduce  the  following  formulae : 

(i).     sinh  {ic  ±v)^  sinh  7C  cosh  v  zt  cosh  2c  sinh  v. 

(2).      cosh  (ic  ±v')=^  cosh  7C  cosh  v  zb  sinh  71  sinh  v. 

,    .  .   ,  .  tanh  71  ±:  tanh  v 

(3).      tanh  (71  zb  Z')  =  — -- — r : — r — . 

^^''  ^  ^        I  zb  tanh  7C  tanh  v 


GONIOMETRIC   AND   HYPERBOLIC   RATIOS. 


63 


(4).  Deduce  these  formulae  also  geometrically  from  the 
constructions  of  Arts.  48,  53,  assuming  for  the  definitions 
of  sinh  2t  and  cosh  u  the  ratios  NP  /  a  and  ON  I  a.  [Burn- 
side:     Messenger  of  Mathe?natics,  vol.  xx,  pp.  145-148.] 

(5).  In  the  figure  of  Art.  53  show  that  the  trapezoids 
SoAP^S^,  S^P^P^S^,  etc.,  are  equal  in  area  to  the  corre- 
sponding triangles  OAP^,  OP^P^,  etc.,  and  consequently 
to  each  other. 

(6).  Show  that  when  the  hyperbolic  sector  OAP  (Art. 
53)  increases  uniformly,  the  corresponding  segment  OS,  laid 
off  on  the  asymptote,  increases  proportionately  to  its  own 
length. 

(7).  Assuming  «=i  in  the  equilateral  hyperbola  of 
Art.  48,  and  that  the  area  of  any  sector  is  ^  ?(,  prove  that 

^^^^  [(sinh  ii)  1 71']=  1.     (Use  the  method  of  Art.  55.) 


55.     An   Approximate    Value    of   Natural    Base. 

We  may  determine  between  what  integers  the  numerical 
value  of  e  must  lie,  by  substituting  their  equivalents  in 
X  and  y  for  the  terms  of  the  inequality : 

Triangle  OAP  >  sector  OAVP  >  triangle  ORS, 
as  represented  in  Fig.  28.  For 
our  present  purpose  it  will  in- 
volve no  loss  of  generality  and 
it  will  simplify  the  computation 
to  assume  OA  =  i ,  so  that  the 
equation  of  the  hyperbola  is 
x^  —  X  =  I  • 
The  sectorial  area  OA  VP,  as 
previously  found  in  Art.  53,  is 
then   4?^,   the  area  of   OAP  is 


R  A 

Fig.  28. 

obviously  \  y,  and  for  that  of  OPS  we  may  write 
^^  OP  X  (ordinate  of  .S"). 


64  GONIOMETRIC    AND    HYPERBOLIC    RATIOS. 

To  determine  OR  and  this  ordinate,  write  the  equations  to 
the  tangent  RS  and  the  Hne  OP^  and  find  the  ordinate  of 
their  intersection,  and  the  intercept  of  the  former  on  the 
j;-axis.     The  results  are  : 


"n 


y 


X 


—  ^-1/2/(^-1), 


the  equation  to  the  tangent  RS,  i  and  rj  being  the  current 
co-ordinates  of  the  Hne ; 

the  equation  to  OP; 

J' 

the  required  intercept  on  the  ^-axis ;  and 


r}=y  2  {x—  l), 

the  ordinate  of  S,  found  by  ehminating  i  from  the  equations 
to  RS  and  OP.      Hence  the  area  ORS  is 

ivOR  =  '^. 
and  the  inequaHty  between  the  areas  takes  the  form 

•^  J' 

Ify  =  I ,  then  w  <  I ,    x  =  1/2,  and 

e"  =  X  -\-y  =  I  H-  1/2  =  2.4+; 
therefore  d"  >  2  .  4. 


*  Had  the  assumption  a  =  i  not  been  made,  this  inequality  would  have  been 

The  equations  for  i^^'and  OP  and  the  expressions  for  OR  and  'y  would  have  been 
correspondingly  changed,  but  the  final  results  would  have  been  the  same  as  those 
given  above. 


GONIOMETRIC   AND    HYPERBOLIC    RATIOS.  65 

1^2^^^  =  I,  then?^>i,  ;r=|  j'=^    and 

therefore  ^  <  3- 

A  nearer  approximation  to  the  value  of  e  is  found  by  other 
methods.     To  nine  decimal  places  it  is  2. 7 1828 1828. 

56.  Agenda.  Logarithmic  Forms  of  Inverse  Hyper- 
bolic Ratios.  It  is  customary  to  represent  by  sinh-^j^, 
cosh— ^  X,  .  .  .  ,  the  arguments  whose  sinh,  cosh,  .  .  .  ,  are 
J',  X,  etc.      Let /^sinh-ij/;  then 

y  =  sinh  /  =  -  (^^  —^0' 
whence,  multiplying  by  ^^  and  re-arranging  terms, 

^2^  —  2>'(?^  —  1=0, 

a  quadratic  equation  in  6^,  the  solution  of  which  gives 

or  /=lnO'±lO^^+i). 

IfjV'  be  real,  the  upper  sign  must  be  chosen ;  for  \/y  -f  i  ^j 
and  e^  is  positive  for  all  real  values  of/  (Art.  23).      Hence 
( I }.      sinh-i_>/  =  In  (j^/  4-  |/y^-|-i). 
Prove  by  similar  methods  the  following  formulae  : 
(2).     cosh-i  ^  =  In  (x  +  ^/x'  —  i). 

(3).     tanh-^^^|ln^^i^' 

I  —  2* 

(4).      coth-^^  =  |ln^i^. 

(5).      sech-^  -^  =  In  ( I  / ;r  +  |/i  /  ji;^  -  I). 

(6).      csch-^j/  =  In  (i  /_;/  -f  |/i/;j/^+i). 

(Cf  Art.  97.) 


CHAPTER    III. 

THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 


XI.     Geometric  Addition  and  Multiplication. 

57.       Classification   of   Magnitudes:     Definitions. 

It  was  pointed  out  in  Art.  2  that  any  one  straight  line 
suffices  for  the  complete  characterization  of  all  so-called 
real  quantities ;  in  fact  the  real  magnitudes  of  algebra  were 
defined  as  lengths  set  off  upon  such  a  line.  But  because, 
in  this  representation,  no  distinction  in  direction  was  neces- 
sary, all  line-segments  were  taken  to  be  real  magnitudes, 
and  comparisons  of  direction  were  made,  by  means  of  the 
principles  of  geometrical  similarity,  for  the  sole  purpose 
of  determining  lengths.  Such  comparisons  will  still  be 
necessary  whenever  the  product  or  quotient  of  two  real 
magnitudes  is  called  for,  but  into  the  real  magnitudes 
themselves  no  element  of  direction  enters ;  their  sole  char- 
acteristics are  length  and  sense,  that  is,  length  and  extension 
forwards  or  backwards. 

If  the  attempt  be  made  to  apply  the  various  algebraic 
processes  to  all  real  magnitudes,  negative  as  well  as  posi- 
tive, another  kind  of  magnitude,  not  yet  considered,  is 
necessarily  introduced.  For  example,  if  x  be  positive,  no 
real  magnitude  can  be  made  to  take  the  place  of  either 
(—  xy'^  or  log,„  ( —  x^\  for  the  square  of  a  real  quantity  is 
always  positive  (Art.  14),  and  the  definition  of  an  exponen- 
tial given  in  Art.  23  precludes  its  ever  assuming  a  negative 
value.      In  order  that  forms  like  these  may  be  admitted  into 


THE  ALGEBRA  OF  COMPLEX  QUANTITIES.      67 

the  category  of  algebraic  quantity,  a  new  kind  of  quantity 
must  therefore  be  defined,  or  more  properly,  a  new  defini- 
tion of  algebraic  quantity  in  general  must  be  given. 

Having  assigned  some  fixed  direction  as  that  in  which 
all  real  quantities  are  to  be  taken,  we  adopt  a  straight  line 
having  this  direction  as  a  line  of  reference,  call  it  the  real 
axis,  and  determine  the  directions  of  all  other  straight  Hnes 
in  the  plane  by  the  angles  they  make  with  this  fixed  one. 
Line-segments  having  directions  other  than  that  of  the  real 
axis  are  the  new  magnitudes  that  now  demand  consideration. 
They  are  called  vectors.  They  have  two  determining  ele- 
ments :  length  and  the  angle  they  make  with  the  real  axis. 

(i).  Its  length,  taken  positively,  is  called  the  tensor  of 
the  magnitude,  and  the  arc- ratio  of  the  angle  it  makes  with 
the  real  axis  is  called  its  amplitude  (or  argic^nent^. 

Classified  and  defined  with  respect  to  amplitude,  the 
magnitudes  themselves  are : 

(ii).       Real,  if  the  amplitude  be  o  or  a  multiple  of  tt  ; 

(iii).  Lnaginary,  if  the  amplitude  be  tt/ 2  or  an  odd 
multiple  of  TT  /  2. 

(iv).      Complex,  for  all  other  values  of  the  amplitude. 

In  general,  therefore,  vectors  in  the  plane  represent 
complex  quantities,  but  in  particular,  when  parallel  to  the 
real  axis  they  represent  real  quantities ;  when  perpendicu- 
lar to  it,  imaginary. 

Any  quantity  is  by  definition  uniquely  determined  by  its 
tensor  and  amplitude,  and  hence : 

(v).  Two  quantities  are  equal  if  their  tensors  and  their 
amplitudes  are  respectively  equal,  the  geometrical  rendering 
of  which  is:  two  magnitudes,  or  vectors,  are  equal  if 
(and  only  If)  they  are  at  once  parallel,  of  the  same  sense, 
and  of  equal  lengths. 


68 


THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 


The  algebra  of  complex  quantities,  like  that  of  real 
quantities,  is  developed  from  the  definitions  of  the  funda- 
mental algebraic  operations:  addition  and  subtraction, 
multiplication  and  division,  exponentiation  and  the  taking 
of  logarithms.  These  operations  applied  to  magnitudes 
represented  by  straight  lines  in  the  plane  are  called  alge- 
braic by  reason  of  their  identity  with  those  of  the  analysis 
of  real  quantities,  but  specifically  geometric,  because  each 
individual  operation  has  its  own  unique  geometrical  config- 
uration. On  the  other  hand,  the  algebraic  processes  applied 
to  real  quantities  may  be  described  as  geometric  addition, 
multiplication,  involution,  etc.,  in  a  straight  line. 


58.  Geometric  Addition.  Regarding  lines  for  our 
present  purpose  as  generated  by  a  moving  point,  the  opera- 
tion of  addition  is  defined  to  mean  that  a  point  P,  free  to 
move  in  any  direction,  is  successively  transferred  forwards  or 
backwards,  that  is,  in  the  positive  or  negative  sense  as 
marked  by  the  signs  +  and  — ,  through  certain  distances 
designated  by  appropriate  symbols  a,  ^,  y  .  .  Thus  the 
sum  4-  a  —  ^  -|-  y,  in  which  a,  ;8,  y  represent  vectors  in 
the  plane  (or  in  space),  joined  to  form  a  zig-zag,  as 
shown  in  the  accompanying  figure,  may  be  read  off  as 
follows,  the  arrow-heads  indicating 
direction  of  motion  forwards:  Move 
forwards  through  distance  a  to  A, 
then  backwards  through  distance 
y3  to  B\  then  forwards  through 
distance  y  to  C\  and  the  result  is 
the  same  as  if  the  motion  had  taken 
place  in  a  direct  line  from  (9  to  C; 
this  fact  is  expressed  in  the  equation 


Fig.  29. 


J^a-[i+y=OC\ 


THE    ALGEBRA   OF   COMPLEX   QUANTITIES. 


69 


If  not  already  contiguous,  the  magnitudes  that  form  the 
terms  of  a  sum,  by  changing  the  positions  of  such  as  require 
it  without  changing  their  direction,  may  be  so  placed  that 
all  the  intermediate  extremities  are  conterminous.  Geo- 
metfic  addition  may  therefore  be  defined  as  follows : 

The  sum  of  two  or  more  magnitudes,  placed  for  the 
purpose  of  addition  so  as  to  form  a  continuous  zig-zag,  is 
the  single  magnitude  that  extends  from  the  initial  to  the 
terminal  extremity  of  the  zig-zag. 

59.     The  Associative  and  Cominutative  Laws  for 

geometric  addition  in  the  plane  are  deduced  as  immediate 
consequences  of  its  definition.  For  in  the  first  place,  the 
ultimate  effect  is  the  same  whether  a  transference  is  made 
from  O  direct  to  B  then  to  C  as 
expressed  by  (a  -f  /5)  -f  y,  in  the 
subjoined  figure,  or  from  O  to  A 
then  direct  from  ^  to  C  as  expressed 
by  a  +  (/3  +  7),  or  from  O  to  A  to 
B  to  C  as  expressed  by  a  +  yS  -f  y; 
hence 

(a  +  ^)+7  =  a-f-(^+T) 
=  a-f /i+y; 

and  in  the  second  place,  by  (v)  of 
57,  in  ABCB>,  a  parallelogram,  AD  =  BC  =  y, 
BC=  AB  =  /?,  and  by  the  definition  of  addition  AB  -f  BC 
=  AC=  AD  +  DC,  whence 

One  or  more  of  the  terms  may  be  negative.  Expressing 
this  fact  by  writing  zt  a,  zh  /?,  d=  y  in  place  of  a,  /j^,  y,  the 
two  resultant  equations  of  the  last  paragraph  become 

rd=a±/?)±y==ba+  (±:^d=y)=±:azi=y5zby, 
±:  y  ±  /i  =  ±  /?  ±  y. 


Art. 


70      THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 

which  express   the  associative   and   commutative  laws   for 
addition  and  subtraction. 

It  is  evident  that  the  principles  of  geometric  addition 
apply  equally  to  vectors  in  space,  or  in  the  plane,  or 
to  segments  of  one  straight  line.  In  particular  algebraic 
addition  (Art.  3)  may  be  described  as  geometric  addition 
in  a  straight  line. 

60.  Geometric  Multiplication.  The  geometric  pro- 
duct of  two  magnitudes  a,  /?,  is  defined  as  a  third  magnitude 
y,  whose  tensor  is  the  algebraic  .product  of  the  tensors  of 
the  factors  and  whose  amplitude  is  the  algebraic  sum  of 
their  amplitudes,  constructed  by  the  rules  for  algebraic 
product  and  sum.      (Arts.   5,   3.) 

If  one  of  the  factors  be  real  and  positive,  the  amplitude  of 
the  other  reappears  unchanged  as  the  amplitude  of  the 
product,  which  is  then  constructed,  by  the  algebraic  rule, 
upon  the  straight  line  that  represents  the  direction  of  the 
complex  factor,  and  it  was  proved  in  Art.  18  that  in  such  a 
construction  an  interchange  of  factors  does  not  change  the 
result.      Hence,  if  a  be  real  and  positive  and  /3  complex, 

In  this  product,  tensor  o{  aX  /3  =  a.  X  tensor  of  fi  by 
definition,  and  if  tensor  of  /?=i,  the  complex  quantity 
aX  ft  appears  as  the  product  of  its  tensor  and  a  unit  factor 
/?,  a  complex  unit,  which  when  applied  as  a  multiplier  to  a 
real  quantity  a,  does  not  change  its  length  but  turns  it  out 

of  the  real  axis  into  the  direction  of  ft. 

Any  such  complex  unit  is  called  a  versoi: 

Let  tsr  stand  for  tensor,  vsr  for  versor ; 

then  every  complex  quantity  a  can  be 

expressed  in  the  form 

a  ^^  tsr  a  X  \'sr  a. 


THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 


71 


This  versor  factor  is  wholly  determined  by  its  amplitude, 
in  terms  of  which  it  is  frequently  useful  to  express  it.  For 
this  purpose  let  i  be  the  versor  whose  amplitude  is  7r/2,  0 
the  amplitude  of  the  complex 
unit:  /5,  OX  the  real  axis,  BiM 
the  perpendicular  to  OX  from 
the  terminal  extremity  of  ft. 
Then  MB  =  sin  (9,  0M=  cos  6, 

and  by  the  rule   of  Q^eometric       _  ^ 

addition.  Fig.  32. 

ft  =  0M-\-  i  X  MB  =  cos  0  +  2  sin  6. 
As  an  abbreviation  for  cos  0  -\-  2' sin  6  it  is  convenient  to  use 
cis  0,  which  may  be  read :  sedor  of  B.  In  this  symbolism, 
the  law  of  geometric  multiplicatioii  (product  of  complex 
quantities,  as  above  defined)  is  expressed  in  the  formula, 
{a  •  cis  </))  X  (^  •  cis  t/^)  =  ^  X  ^  •  cis  (<^  +  ./^). 

It  is  obvious  that  algebraic  multiplication,  described  in 
Art.  5,  is  a  particular  form  of  geometric  multiplication, 
being  geometric  multiplication  in  a  straight  line. 

61.  Conjugate  and  Reciprocal.  If  in  the  last  equa- 
tion b  =  a  and  i/^  =  —  <j>,  it  becomes 

{a  '  cis  </))  X  (^  •  cis  [—  <35)] )  =  <^  X  ^  *  cis  o  =  ^^ 

The  factors  of  this  product,  a  cis  <^ 
and  a  cis  ( —  <^),  are  said  to  be  conju- 
gate to  one  another,  and  we  have  the 
rule : 

The  product  of  two  conjugate  com- 
plex quantities  is  equal  to  the  sqiiare 
of  their  tensor. 

If  this  tensor  be  i ,  the  product  re- 
duces to 
cis  <^  '  cis  (—</))  =  i; 


72      THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 

that  is,  by  virtue   of  the  definition  of  Art.   6,   cis  <f>  and 
cis  ( —  <f>)  are  reciprocal  to  each  other  and  we  may  write 

/  cis  cf>  =  cis  ( —  <^). 

In  Hke  manner,  since  it  is  now  evident  that 

(^  •  cis  <^)  X  (/<2  •  /  cis  <^)  =  <2  /<2  •  cis  o=  I 
/  (a  •  cis  <f>)  =^  /  a  '  /  cis  <i>. 

If  now  in  the  formula  expressing  the  law  of  multiplication 
we  write  /  d  and  —  i/^  for  /5  and  iff  respectively,  we  have,  as 
the  law  of  geometric  division, 

(a  '  cis  <^)  /  (d  '  cisij/')  =  a  /  d  '  cis  (^  —  i/^). 

62.  Agenda.  Properties  of  cis  <^.  If  71  be  an  integer 
prove : 

( I ).      (a'  cis  0)"  =  a''  '  cis  n  <p. 

(2).      cis  (<^  zh  2  72  tt)  =  cis  <^. 

(3).      cis  (<^  ±  \_2  71  -\-  1']  tt)  =  —  cis  (p. 

(4).      cis  (<^  -f  [2  72  ±  ^]  tt)  =  zlz  /  cis  <^. 

(5).      cis  (<^  +  [2  ?2  d=  I ]  tt)  =  =F  /  cis  <^. 

(6).  Show  that  the  ratio  of  two  complex  quantities 
having  the  same  amplitude,  or  amplitudes  that  differ  by 
dz  27r,  is  a  real  quantity. 

(7).  Show  that  the  ratio  of  two  complex  quantities 
having  amplitudes  that  differ  by  ±  ^tt  is  a  purely  imag- 
inary quantity. 

63.  The  Imaginary  Unit.  By  definition  (iii)  of 
Art.  57  cis  7r/2  =  z  is  an  imaginary  having  a  unit  tensor; 
it  is  therefore  called  the  imaginary  unit.  Its  integral  powers 
form  a  closed  cycle  of  values ;  thus  : 

.     1"      .  .  .     "iTT  .      . 

2  =  CIS-,  Z^=:=ClS7r=  —  I,  z^=ciS— -=  —  Z,  Z''=C1S  27r=  I, 


THE  ALGEBRA  OF  COMPLEX  QUANTITIES.      73 

and  the  higher  powers  of  i  repeat  these  values  In  succession ; 
that  is,  if  11  be  an  integer, 

and  these  are  the  only  values  the  integral  powers  of  i  can 
acquire. 

64.     The  Associative   and   Commutative    Laws. 

Let  a,  b,  c  be  the  tensors,  ^,  ip,  x  ^^^  amplitudes  of  a,  /3,  y 
respectively;  that  is, 

a  =  <2*cis^,    (3  =  d  '  CIS  if/,     y=c'cisX' 

Then,  by  the  law  of  geometric  multiplication, 

aX  (/3  X  y)  =  (^  •  cis <A)  X  ([<5-cisiA]  X  [^'cisx]) 
=  (a-cis<A)  X(bXc'cis[i}/-\-x']) 
=  aX(bXc)  •  CIS  (ct> -i- [^ -{- x])  ; 

and  by  the  same  process, 

(a  X  ^)  X  y  =  («  X  ^)  X  ^  •  cis  ([<A  +  ^]  -f  x)- 

But,  by  the  rules  of  algebraic  multiplication  and  addition, 

aX(^X0  =  («X^)X^and<^4-('A  +  x)  =  (<^  +  'A)+x; 
.••     aX(^X7)  =  (aXi8)Xy. 

which  is  the  associative  law  for  multiplication. 

And  again,  by  the  law  of  geometric  multiplication, 

a  X  y^  =  (^  •  cis  <^)  X  (<5  •  cis  «/r) 

=  ^X^-cis(<^  +  'A), 
and  similarly  /5Xa  =  <^X«*cIs(«/^X  <^). 

But,  by  the  rules  of  algebraic  multiplication  and  addition, 

aX  b  =  d  X  a  and  <f>  -{-  ij/ =  ij/ -{-  <i> ; 
.-.     aXf^  =  /3Xo., 

which  is  the  commutative  law  for  multiplication. 


74 


THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 


The  letters  here  involved  may  obviously  represent 
either  direct  factors  or  reciprocals,  and  the  sign  X  may 
be  replaced  at  pleasure  by  the  sign  /,  without  affecting  the 
proof  here  given  (see  Art.  6i).  Hence,  as  in  Arts.  32,  33, 
with  real  magnitudes,  so  with  complex  quantities  the  asso- 
ciative and  commutative  laws  for  multiplication  and  division 
have  their  full  expression  in  the  formulae, 


65.  The  Distributive  Law.  From  the  definitions 
of  geometric  addition  and  multipHcation  (Arts.  58,  60)  the 
law  of  distribution  for  complex  quantities  is  an  easy  conse- 
quence. The  construc- 
tions of  the  subjoined 
figure,  in  which  a,  ft 
and  VI  '  cis  0  represent 
complex  quantities  will 
bring  this  law  in  direct 
evidence.  The  opera- 
tion m'cisO  changes 
OA  into  OA',  AB= 
A^  mto  A E=A^\ 
and  OB  Into  0B\  that 
is,  turns  each  side  of  the 
triangle  OAB  through 
the  angle  AOA'  and  changes  its  length  in  the  ratio  of 
w  to  I,  producing  the  similar  triangle  OA' B\  in  which 

0~A'  =  VI  •  cis  ^  X  a, 

A^'  =  VI '  cis  e  X  A 

Ub'  =  m  •  cis  ^  X  (a  +  /?). 
But,  by  the  rule  of  geometric  addition, 


Fig-  34' 


THE  ALGEBRA  OF  COMPLEX  QUANTITIES.      75 

.' .     m'  ds9x{p.-\-  ft)=  VI  •  cis  (9  X  a  +  ?;?  •  cis ^  X  /?. 

This  demonstration  i.s  in  no  way  disturbed  by  the  intro- 
duction of  negative  and  reciprocal  signs.  The  last  equation 
above  written  is,  in  fact,  the  first  equation  of  page  37,  and  the 
subsequent  equations  of  Arts.  20,  21  and  their  proofs  remain 
intact  when  for  the  real  quantities  a,  b,  c,  d,  etc. ,  complex 
quantities  are  substituted.      Hence,  writing  w  •  cis  ^  =  7, 

(±  a  ±  /?)  >5.  (±:  y)  =  +  (±  a)  >5^  (+  y)  +  (±  /?)  >5^  (zh  y). 

Here,  as  in  Arts.  20,  21,  the  sign  X  is  distributive  over  two 
or  more  terms  that  follow  it,  but  not  so  the  sign  /. 

66.  Argand's  Diagram.  It  is  obvious  from  its  defi- 
nition as  here  given  (Art.  57)  that  to  every  complex 
quantity  there  corresponds  in  the  plane  a  unique  geo- 
metrical figure  which  completely  characterizes  it.  This 
figure  is  known  as  Argand's  diagram,*  and  consists  of  the 

real  axis  OX  with  reference  to 
which  the  arc-ratio  9  is  esti- 
mated, the  imaginary  axis  O  V 
perpendicular  to  OX,  the 
directed  line  OP  that  repre- 
sents the  complex  quantity  and 
fis--S5-  the  perpendicular  PA  from  P 

to  OX.  The  axes  OX  and  O  V  are  supposed  to  be  fixed 
in  position  and  direction  for  all  quantities.  Any  point  P 
in  the  plane  then  determines  one  and  only  one  complex 
quantity  and  one  set  of  line-segments  OA,  AP,  OP, 
different  from  every  other  set. 


*  First  constructed  for  this  purpose  by  Argand  :  Essai  snr  une  manUre  de 
representer  les  qiiantites  imaginab es  dans  les  constructions  geonietriques;  Paris, 
1806.    Translated  by  Prof.  A.  S.  Hardy..  New  York,  1881. 


76      THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 

If  OA  =  X  and  AF=^y,  the  complex  quantity  appears  in 
the  form  x-\-iy,  and  if  OP=a  and  arc-ratio  of  A0P=6, 
the  relations 

X  =  a  cos  0,  y  =  a  sin  0, 

a''  =  x^  -\-y%   t2in9^=y  I X 

are  directly  evident  from  the  figure  and  we  have 

X  ^  iy=^a(^cos6 -^  ismO), 
XsY{x-\-iy)==-^\/x'-\-y\ 
vsr  {x  +  iy')  =  cos  6  -\- z  sin  6, 
amp  (.r  -f-  {y)  =  arc-ratio  whose  tangent  isy  /  x. 

In  analysis  the  complex  quantity  most  frequently  presents 
itself  either  expHcitly  in  the  form  x  -\-  iy,  or  impHcitly  in 
some  operation  out  of  which  this  form  issues. 

67.  Agenda.  Multiplication,  Division  and  Construc- 
tion of  Complex  Quantities.     Prove  the  following  : 

( I ).      (^a-\-  ib)  X  {x  -{-  I'y)  =^  ax  —  by  -\-  i(^ay  ^  bx). 

(2).      ia+ib)  I  (.  +  iy)^^^  +  ^->;+ff"  -  "-^X 

(3).      {a  +  iby  +  («  -  iby=2  {a'  +  b')  —  I2a'b\ 
la  -\-  ib\-      la  —  iby  ^iab 


.    .       la-\-  toy      la  —  zby ^.zab 

U;-    \^^r7^;  -  \«qr^/  —  («=  +  b^y 
(5).    TT7  +  T=r7  =  3. 


I         X-  — y^  —  2ixy 

(x-^iyy==     {x^-vyy 

X  -\-  iy      x^  —  3.ry^  -f  ^  iz^y  — "JVO 

{x-^ylif  —  ' ix~  -{-yy 


(6). 

(7). 

(8).      (_  ^  +  /I  ,/3)^=  _  I  _  /1 1/3. 


THE  ALGEBRA   OF   COMPLEX   QUANTITIES. 


77 


(9).     (-i  +  ^iV3y=(-i-^iV3r=l^ 
(lo).   \/x  H-  ly 

(II).    [±:  (I  +  0  /  t/2]^=  [±  (I  -  0  /  1/2]^=  -  I. 

(12).  Write  down  the  expression  for  tensor  in  each  of 
the  above  examples. 

(13).  Prove,  by  the  aid  of  Argand's  diagram  (Art.  66), 
that  the  tensor  of  the  sum  of  two  or  more  complex  quan- 
tities cannot  be  greater  than  the  sum  of  their  tensors;  that  is, 

tsr(a-f  ;8)<tsra-{-tsri8. 

(14).     By  definition  (Art.  60), 

tsr  (  a  >5/  )8  )  =  tsr  a  >^  tsr  /8, 
and  amp  (  a  >^  yS  )  =  amp  a  dz  amp  /?, 

and  hence  no  proofs  of  these  properties  are  called  for. 

Construct  the  following,  applying  for  the  purpose  the 
rules  of  algebraic  and  geometric  addition  and  multiplication 
(Arts.  3,  5,  58,  60): 

(15).  («  +  /^)  +  (:t-H-iy). 

(16).  {a-\-ib)-{x-\-iy). 

(17).  (a^2b)X{x^iy), 

(18).  (^ci-^ib)l{x  +  iy). 

(19).  (^  +  ^»^ 

(20).  I  /  (-r  +  iy)- 

(21 ).  (a  •  cis  i>)  X  (b'  cis  tp), 

(22).  b  /  (a  •  ciscf>). 

(23).  (a-c[s<f>)\ 

(24).  (a  '  cis  (f>)  /  (b  '  cis  ^)^ 


78      THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 

XII.     Exponentials  and  Logarithms.* 

68.  Definitions.  In  a  circle  whose  radius  is  unity, 
OT  is  assumed  to  have  a  fixed  direction,  its  angle  with  the 
realaxis=/(97',  (Fig.  36),  OR  is  supposed  to  turn  about 


Fig.  36. 


O  with  a  constant  speed,  Q  to  move  with  a  constant  speed 
along  any  line,  as  ES,  in  the  plane,  P  along  OR  with  a 
speed  proportional  to  its  distance  from  O.     Let 


*  The  theory  of  logarithms  and  exponentials,  as  here  formulated,  was  the  sub- 
ject-matter of  a  paper  by  the  author,  entitled  "  The  Classification  of  Logarithmic 
Systems,' '  read  before  the  New  York  Mathematical  Society  in  October,  1891,  and 
subsequently  published  in  the  American  Journal  of  Mathematics,  Vol.  XIV,  pp. 
187-194.  It  was  further  discussed  by  Professor  Haskell  and  by  the  author  in  two 
notes  in  the  Bulletin  of  the  New  York  Mathematical  Society,  Vol.  II,  pp.  164-170. 


THE    ALGEBRA    OF   COMPLEX    QUANTITIES.  79 

Speed  of  P  in  OR  at  A^X,    arc-ratio  of  /OR  ^  0, 

speed  of  Q  in  BS  ^/x.,    arc-ratio  of  /DS^cf>, 

speed  of  R  in  /RS          ^  w,    arc-ratio  of /O  7"^  /3, 

OP,   OQ~p,  q,    OM,  MP^x,y, 

ON,  NQ=u,  V,    0N\  N'Q^u\  v\ 

OC^c,  JA^a  —  (a  possible  multiple  of  27r), 

O  T^  cos  (i  Ar  i  sin  /?  ^  cis  /?,    ?;^  cis  /?  ^  k, 

i^'Z'  will  be  called  the  modular  line,  and  OF,  drawn 
through  the  origin  perpendicular  to  ET,  will  be  called  the 
modular  7iormal. 

In  all  logarithmic  systems  the  relation 

a)/A.=  tan(c^  —  /5) 

is  assumed  to  exist,  and  this,  together  with  the  equation 
jw.  /  1,/A.^  -f  w^  =  m,  by  elimination  of  co,  gives,  as  a  second 
expression  for  ?;z, 

ni  ==  fji  /  \'  cos  ((^  —  /?). 

Let  the  values  of  w  and  /3  be  assigned,  and  the  path 
and  speed  of  Q  determined,  by  fixing  the  angle  JDS, 
the  position  of  the  point  Cand  the  value  of  )u,.  The  value 
of  A  is  then  completely  determined  through  the  equation 
m  =  iL  I  X'  cos  (</)  —  /5),  and  the  value  of  w  by  the  previous 
equation  to  =  A  tan  (<^  —  f3).  Thus  the  curve  upon  which 
P  moves,  when  Q  moves  upon  a  known  straight  line,  is 
given  its  definite  form  by  the  values  assigned  to  m  and  /?, 
which  are  therefore  the  two  independent  determining  factors 
in  any  logarithmic  system. 

But  it  is  still  unknown  whether,  when  the  position  of  Q 
is  assigned,  P  is  far  or  near,  and  in  order  to  completely 
define  the  position  of  P  relatively  to  that  of  Q  let  it  be 


8o      THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 

agreed  that  when  Q  traverses  some  specified  distance  along 
its  straight  path  P  shall  pass  through  a  definite  portion  of 
its  curved  path;  whence  it  will  then  follow  that  to  every 
position  of  Q  corresponds  a  known  unique  and  deter- 
minate position  of  P. 

A  convenient  assumption  for  this  purpose  is  found  to  be 
that  while  Q  is  passing  from  the  modular  line  to  the  modular 
normal  {E  to  C),  P  shall  go  from  a  point  on  the  real  axis 
to  the  circumference  of  the  unit  circle,  upon  a  segment  of 
its  path,  which,  if  necessary,  may  wind  about  the  origin 
one  or  more  times ;  6  increasing  meanwhile  from  o  to  JA 
or  from  o  to  JA  plus  a  multiple  of  27r.  This  assumption 
determines  C  and  A  as  corresponding  points  and  fixes  A  as 
a  definite  point  on  the  circumference  of  the  unit  circle  (see 
Art.  72). 

Having  thus  set  up  a  unique  or  one-to-one  correspond- 
ence between  the  positions  of  P  and  Q  in  their  respective 
paths,  we  define  the  terms  modulus,  base,  exponential  and 
logarithm  as  follows  : 

(i).  The  modulus  is  the  product  of  the  two  inde- 
pendent quantities  mand  cis  /?;  that  is,  if  k^  modulus, 


K  =  fx/i/\^-\-o}''  (cos /3 -^  i s'm  ft) 
=  fji  I  X'  cos(<^  —  /3)  •  (cos/3-f  ism  ft). 

(ii).  The  base  is  the  value  that  OP^  or  x  -\-  iy,  assumes 
at  the  instant  when  OQ  becomes  i,  that  is,  when  Q  passes 
through  the  point  J  as  it  moves  along  some  line  that  inter- 
sects the  unit  circle  at  J.  In  general  B  will  stand  for  base 
corresponding  to  modulus  k. 

(iii).  OP  is  the  exponeyitial  of  OQ,  either  with  respect 
to  the  modulus  /c,  as  expressed  by  the  identity 

X  -f  ?>'=  exp^  {n  4-  iv). 


THE  ALGEBRA  OF  COMPLEX  QUANTITIES.       8l 

or  with  respect  to  the  base  B,  as  expressed  by  the  identity 

(iv).  Inversely,  OQ  is  the  logarithm  of  OP,  either 
with  respect  to  the  modulus  /c,  as  expressed  by  the  identity 

or  with  respect  to  the  base  B^  as  expressed  by  the  identity 
u  4-  iv  ^  -^log  (  ^*  +  y')- 

69.  Exponential  of  o,  i,  and  Logarithm  of  i,  B. 

If  the  path  of  Q  pass  through  the  origin,  the  points  E  and 
C  will  coincide  at  O  and  the  path  of /^  will  cross  both  the  cir- 
cumference of  the  unit  circle  and  the  real  axis  at  J.     Hence 

y=^o  and  Jf  =  i ,  when  7^  =  z;  =  o, 

to  which  correspond  the  convenient  relations 

exp^o  =  ^°=  I, 
and  log;^  I  =^log  I  =0. 

Here  also,  as  in  Art.  23,  because  w=^\  when  z=^  B, 

log^ B  =  I,  and  exp^  i  =  B^  =  B. 

70.  Classification  of  Systems.  The  special  value 
zero  for  the  modular  angle  /OT  eliminates  the  imaginary 
term  from  the  modulus  and  introduces  the  ordinary  system 
of  logarithms,  with  a  real  modulus.  A  system  is  called 
g-onic,  or  a-gonic,  according  as  its  modulus  does  or  does 
not  involve  the  angular  element  /3. 

The  geometrical  representation  of  agonic  systems  is 
obtained  from  Fig.  36  by  turning  the  rigidly  connected 
group  of  lines  EN\  EQ  and  OF,  together  with  the  speci- 
fied points  upon  them,  around  the  origin  as  a  fixed  centre. 


82      THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 

backwards  through  the  angle  JOT^  so  that  7",  S  fall  into  the 
positions  /,  S\  D  and  E  then  coincide  upon  JD,  OF 
becomes  perpendicular  to  yZ>,  ^  —  ,3  remains  unchanged 
in  value  but  merges  into  <^,  and  the  new  modulus  becomes 
ft  /  |/  V  -f-  to^,  or  /A  /  A.  •  cos  <^,  its  former  value  with  the  factor 
cis  /3  omitted ;  while  no  change  in  X,  /x  and  a>  need  take 
place.  Thus  the  path  of  P  remains  intact,  and  the  new 
Q  moves  with  its  former  speed  in  a  straight  line  passing 
through  S'  and  through  a  point  on  OJ2X  a  distance  to  the 
left  of  O  equal  to  OE. 

Hence  the  values  of  z  in  the  two  systems  are  identical, 
while  w,  of  the  original  gonic  system,  in  virtue  of  the  back- 
ward rotation  through  the  angle  JOT^  is  transferred  to  the 
new  or  agonic  system,  by  being  multiplied  by  cis  (—  /3),  so 
that  the  original  w  and  its  transformed  value,  here  denoted 
by  w\  bear  to  one  another  the  relation 

w  =  zv'  cis  /3. 

The  agonic  system  above  described  obviously  has  for 
its  equations  of  definition  (Art.  68) 

w'=\og„,z,    2  =  5"^', 

in  which  in  =  jx  j  \-  cos  <^  and  b  is  the  value  of  2  for  which 
\og,^2=  I.  The  formula  connecting  logarithms  in  the  two 
systems  therefore  is 

^^Sk  ^  =  cis  /3  •  \og„i2.  [k  =  7n  cis  i3.  ] 

Finally,  if  in  this  equation  2  =  B,  the  resulting  relation 
between  /3  and  B  is 

cis(— /3)  =  log„,^, 

or  in  the  inverse  form  it  is 

B  =  exp,,, ;  cis  (-  /3)  1=  ^^is(-^). 


THE    ALGEBRA   OF   COMPLEX   QUANTITIES.  83 

71.  Special  Constructions.  A  further  specializa- 
tion is  obtained  by  making  </,  =  /3  ==  o.  Q  then  moves  upon 
a  line  parallel  to  Of,  and  since  tan  (<^  —  ,3)  is  now  zero,  w 
is  also  zero,  i?  remains  fixed  at  A,  and  F  moves  upon  a 
straight  line  passing  through  O  and  A,  This  is  a  con- 
venient construction  for  logarithms  and  exponentials  of 
complex  quantities  with  respect  to  a  real  modulus.  The 
base  is  here  also  real  (Art.  70). 

In  particular,  if  Q  moves  upon  the  real  axis  itself,  A 
coincides  with  /,  F  also  moves  upon  the  real  axis,  and  the 
resulting  construction  is  that  described  in  Art.  23  for  loga- 
rithms of  real  quantities. 

Returning  to  the  general  case  in  which  /3  is  not  zero,  we 
are  at  liberty,  by  our  original  hypothesis  concerning  the 
motion  of  Q,  to  permit  Q  to  move  upon  any  straight  line 
in  the  plane,  provided  we  assign  to  P  such  a  motion  as 
shall  consist  with  the  definition  that  OQ  shall  be  the  loga- 
rithm, to  modulus  K,  of  OF.  Accordingly,  let  Q  be  sup- 
posed to  move  parallel  to  EJV' -,  <^  _  /3  is  then  zero,  <o  is 
likewise  zero,  and  F  moves  upon  a  straight  line  passing 
through  O  and  A.  Or  again,  let  Q  move  parallel  to  OF-, 
cf>—  13  is  then  tt  /  2,  A.  is  zero,  and  F  moves  in  a  circum- 
ference concentric  with  the  unit  circle.  Such  constructions 
are  possible  to  every  logarithmic  system  and  enable  us  to 
simplify  tbe  graphical  representation  of  the  relative  motions 
ofF  and  Q.'-^ 

Expressions  of  the  form  log^;i:,  for  which  no  interpreta- 
tion could  be  found  in  terms  of  real  quantities  unless  x  were 
real  and  positive  (Art.  57),  will  henceforth  be  susceptible 
of  definite  geometrical  representation  for  all  possible  values 
of  ;r.     See  the  examples  of  Art.  86. 

*  We  might,  in  fact,  propose  to  assign,  as  the  path  of  Pin  the  first  instance, 
any  straight  line  passing  through  the  origin,  define  OPas  the  exponential  of  O^). 
and  thtn  determine  the  modulus  by  the  appropriate  auxiliary  construction. 


84  THE    ALGEBRA    OF    COMPLEX    QUANTITIES. 

72.  Relative   Positions   of  A  and    C  in  Fig.  36. 

We  have  by  definition, 

c^OC,    a^JA  -f  a  possible  multiple  of  27r, 
vi^ix  I  \'  cos  (<^  —  /3), 
w  =  Atan  (<^  —  /3). 

The  product  of  the  last  two  of  these  equations  gives 

m  w  =  /xsin  (<^  —  /3), 

But  to  and  /x  sin  (<^  —  /3)  are  the  rates  of  change  of  6  and  v' 

respectively,    and   ^  =  0,    v^=o   are   simultaneous    values 

(Art.  68), 

.  • .     mO^=v'  \ 

and  since  ^  =  a,  v'=-  c  are  also  simultaneous  values  (Art.  68) , 

Thus  A  has  always  a  definite  position  depending  upon  the 
modulus  and  the  distance  from  the  origin  at  which  Q  crosses 
the  modular  normal. 

Since  a  is  the  length  of  arc  over  which  R  passes  while  Q 
passes  from  E  to  C,  it  is  evident  that  when  c  /  7n  lies 
between  2kTr  and  2  (/&  -f-  i )  tt,  say 

c  I  in  =  2  kir  ^  (< 2  tt) 

where  k  is  an  integer,  the  part  of  i^'s  path  that  corresponds 
to  EC  encircles  the  origin  k  times  before  it  intersects  the 
circumference  of  the  unit  circle,  and  the  point  upon  the  real 
axis  that  corresponds  to  E  is  its  {k  -|-  i)^^^  intersection  with 
the  path  of  F,  counting  from  /  to  the  left. 

73.  The    Exponential  Formula.      If  ^  =  arc-ratio 

of  MOP, 

^=/cis^  =  ^"  +  '^. 

It  is  required  to  find/  and  0  as  functions  of  u  and  v. 


THE   ALGEBRA    OF   COMPLEX   QUANTITIES.  85 

Since  the  speed  of  yV^'' In  6^7" Is  ^t  cos  (cf>  —  /3)  and  that  of  P 
in  OJ?  is  Xp,  and  since  by  definition  m:=  fx  /  \  '  cos  (<^  —  /3), 
the  relation  between  OJV\  =^  ?/  and  OF,  =/),  two  real 
quantities,  is  that  of  an  exponential  to  its  logarithm,  with 
respect  to  the  modulus  m  (Art.  23);  that  is,  If  ^=  base 
corresponding  to  modulus  m, 

t/=  log,,,  p,  and  p  =  ^"' ; 

and  in  Art.  72  it  was  shown  that 


Hence  1^"+'-^=  d"'cis~  • 

But  by  considering  the  projections  of  u,  v  upon  u\  v\  in 
Fig.  36,  we  easily  discover  that 


2t  cos  [i  -\-  vsin  /3, 
V  cos  /3  —  7ism  [i 


^u  +  iv  —-  ^wcos^+z^sin^ff  •  q^^ 


V  COS  /3  —  u  sin 


ni 

Or  since  b=^e'^""^  (Art.  24),  this  formula  may  be  written 
„   ,  ,        o  ,     •    o  ^  .2/  cos  /3  —  2^  sin  /3 

When  /3  =  o,  it  becomes 


)u+iv  ^  ^u  (cos  -  +  /  sin  -  ) ' 


and  when  m  =  i  and  therefore  b=^e,  It  assumes  the  more 
special  form 

^ti+tv  ^  ^u  (cos  v^i  sin  z'), 

an  equation  due  to  Euler.* 


*  Init  oductio  ni  Ai/a/ysi/i  hifinitorum,  cd.  Nov.,  1797,  Lib.  I,  p.  104. 


86  THE    ALGEBRA    OF    COMPLEX    QUANTITIES. 

74.  Demoivre's  Theorem.  When  zc  =  o,  Euler's 
formula  becomes 

giv  __  cis  1)^ 

and  by  involution 

^inV    -_   ^(.jg   y^n  __  (xjg   ^^^ 

or  (cos  V  ^  i  sin  z^)"  =  cos  7iv  -f-  ^  sin  wz^ 

for  all  real  values  of  n.  This  equation  is  known  as  De- 
moivre's theorem.^ 

75.  Relations  between  Base  and  Modulus.     Let 

the  lines  EN\  EQ  and  OF  be  regarded  for  the  moment  as 
rigidly  connected  with  one  another  and  be  turned  con- 
jointly in  the  plane  about  the  fixed  point  O  through  an 
arbitrary  angle,  whose  arc-ratio  may  here  be  denoted  by  y. 
ON^  in  the  new  position  thus  given  it,  then  forms  with 
0/  an  angle  whose  arc-ratio  is  y3  -j-  y,  the  modulus  k, 
=  m  cis  /3,  by  virtue  of  this  change,  becomes 

m  cis  ( /3  -f  y)  =  ffi  cis  /3  •  cis  y  =  a;  •  cis  y, 

and  since  OQ,  in  common  with  the  other  lines  with  which 
it  is  connected,  is  turned  about  O  through  the  angle  of  arc- 
ratio  y,  w  is  hereby  transformed  into  zt^cisy;  while  the 
locus  of  F  is  in  no  way  disturbed  by  any  of  these  changes. 
Hence 

ze;  cis  y  =  cis  y  '  log^  z  =  log^  cisy-S". 

In  a  second  transformation,  let  the  motion  of  F  still 
remain  undisturbed,  while  the  speed  of  Q  is  changed  from 
fx  to  nfj.  (?i  =  a  real  quantity).  By  this  change  the  modulus 
Kcisy  becomes  ?ZKcisy,  the  distance  of  Q  from  the  origin 
becomes  ?z^  instead  of  ^,  and  zc^  cis  y  is  transformed  into 
71^  cis  y.      Hence,  writing  -pi  cis  y  =  v,  we  have 


Demoivre  :  Miscellanea  Analytica  (Lond.,  1730),  p   i. 


THE   ALGEBRA   OF    COMPLEX   QUANTITIES.  87 

VW=V  \0g^2  =  \0g.,^Z, 

in  which  v  is  any  complex  quantity,  and  we  may  reiterate 
for  gonic  systems  of  logarithms  the  first  proposition  of 
Art.  24  : 

(i).  To  viidtiply  the  fnodtdiis  of  a  logaritnni  by  any 
quantity  has  the  effect  of  multiplying  the  logarithm  itself  by 
the  same  quantity. 

Corresponding  to  this  equation  connecting  logarithms 
in  two  systems  whose  moduli  are  k  and  vk,  the  inverse,  or 
exponential  relation  is 


expy;^  vw  =  exp^  w 


-^Pk(-'<^)- 


Let  C  be  the  base  in  the  system  whose  modulus  is  vk  ;  then 
the  following  equations  co-exist : 

2e;=log^2',    ^^  =  exp^  2X^  =  j5^, 
vw  =  log^K  z,    2  =  exp;,^  vza  =  C^, 

in  which  are  involved,  as  simultaneous  values  of  w  and  2, 

zv-=  i^      when  z^=  B, 

w=  ijv,  when  z=C.        (Art.  68  (ii).) 

These  pairs  of  values,  substituted  successively  in  the  fourth 
and  second  of  the  previous  group  of  equations,  give,  as  the 
relations  connecting  B,  C  and  v^ 

B=C\    C=B"\ 

Hence  we  may  reiterate  for  gonic  systems  of  logarithms  the 
second  proposition  of  Art.  24  : 

(ii).     If  the  7nodulus  be  changed  from  k  to  vk,  the  corre- 
sponding base  is  cha7igcd from  B  to  B^'  '^. 


88      THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 

The  third  proposition  of  Art.  24  is  a  corollary  of  this 
second;  for  if  the  modulus  be  changed  from  k  to  i,  the 
base  is  thereby  changed  from  B  to  B^  \  that  is, 

exp^  K  =  B*^  =  ^ ;  or 

(iii).  The  exponential  of  any  quantity  with  respect  to 
itself  as  a  modulus  is  equal  to  7iatural  base. 

Finally,  if  B  be  substituted  for  2  in  the  equation 
\og^z=K\n2,  which  is  a  special  case  of  the  formula  of 
proposition  (i),  the  resulting  relations  between  k  and  B 
are 

K=il\nB,    B  =  e^'^, 

and,  in  terms  of  its  base  and  of  natural  logarithms,  the 
logarithm  to  modulus  k  is 

log^  2=-.\x\.2    In B. 

76.  The  La\v  of  Involution.  By  virtue  of  propo- 
sition (i)  of  last  Article  we  have  in  general 

exp^K  w  =  exp^  cv  !  /, 
and 

exp^  iW  =  exp^ tw. 
Hence 

exp^  /ze;  =  exp^  /  ^  ze;  ==  exp^  .^  t  =  exp^  j^^i. 

Otherwise  expressed,  since  changing  k  into  k  / 1,  or  k  /  w, 
changes  B  into  B^,  or  B'^,  the  statement  contained  in  this 
set  of  equations  is  that 

B^'^ =(B^y''  =  (B'^y. 

Obviously  t  or  za,  or  both,  may  be  replaced  by  their  re- 
ciprocals in  this  formula,  and  the  law  of  involution,  more 
completely  stated,  is 

(B^^-)>^'^=(B>-y'^)^'.         (Cf  Art.  25.) 


THE   ALGEBRA    OF    COMPLEX    QUANTITIES.  89 

77.  The  Law  of  Metathesis.  Let  2=^J3^;  then 
by  the  law  of  involution 

and  to  these  there  correspond  the  inverse  relations 
ze;  =  log^2-, 

whence  the  law  of  metathesis, 

t\o%^^z  =  \Qg^zK  (Cf.  Art.  26.) 

Also,  by  changing  the  modulus  (Art.  75)  we  may  write 

/  loge„  5- =  ze/ log/ -2".  (Cf.  Art.  36.) 

78.  The  Law  of  Indices.  Let  w  and  /  be  any  two 
complex  quantities,  w^ii  -\-  iv,  t=r  -]-  is,  in  which  ?^,  v,  r 
and  J  are  real.     By  the  exponential  formula  (Art,  73), 

„         ,        ff  ,     •  tf  •   ^  cos  /3  —  u  sin  /3 

VI 

r>t       7 ^^^.# a.. <.;„/?    •  ^cos /3  —  rsiny3 

VI 

in  which  vi  and  b  correspond  to  one  another  as  modulus 
and  base  respectively  in  an  agonic  system  of  logarithms, 
and  are  both  real.  Hence,  by  the  laws  of  geometric  multi- 
phcation  and  division  (Arts.  60,  61), 

„    ^,    „,       J ,   A-  \      p  ,  ,   4-  X  •    o    .    (z'=t5)cos/3  — (?<;z!=r)sin,3 
'^  m 

But,  by  the  laws  of  geometric  addition  and  subtraction 
(Art.  58), 


90      THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 

{u  ±r)^i{v±s')  =  {2C-\-  iv)  zh  (r  +  is) 

Thus   the   law  of  Indices  obtains  for  complex  quantities. 
(Cf.  Art.  27.) 

79.  The  Addition  Theorem.  Operating  upon  both 
sides  of  the  equation  last  written  with  log^,  we  have 

log^(^->^^0=^±A 

which,   if  B"'^  =  z  and  B^  =  z\   and  therefore   ^-^log^z^^ 
and  2-'=  log^  /,  becomes 

which   is   the   addition    theorem    for   complex    quantities. 
(Cf  Art.  28.) 

80.  The  Logarithmic  Spiral.  The  locus  of  /*,  in 
Fig.  36  of  Art.  68,  is  obviously  a  spiral.  Its  polar  equation 
may  be  obtained  by  considering  the  rates  of  change  o(  p 
and  0,  which  are  respectively  Xp  and  <ur=  A.tan  (<^  — /3). 
By  the  definition  of  a  logarithm  for  real  quantities  (Art.  23), 
6  is  here  the  logarithm  of/  with  respect  to  the  modulus 
A  tan  ((^  —  ,3)  /  A  =  tan  (<^  —  3 ) ;  or  in  equivalent  terms, 

^  =  tan(<A  — ,3)  -In/, 

which  is  the  equation  sought. 

This  locus  is  called  the  logariihmic  spiral.  It  is  obvious 
from  the  definition  of  the  motion  of  the  point  that  generates 
this  curve  (Art.  68),  that  the  spiral  encircles  the  origin  an 
infinite  number  of  times,  coming  nearer  to  it  with  every 
return,  and  that  it  likewise  passes  around  the  circumference 
of  the  unit  circle  an  infinite  number  of  times,  going  con- 
tinually farther  away  from  it. 


THE   ALGEBRA   OF    COMPLEX    QUANTITIES.  9I 

8i.  Periodicity  of  Exponentials.  From  its  char- 
acter as  an  operator  that  turns  any  line  in  the  plane  through 
the  angle  whose  arc-ratio  is  6,  it  is  evident  that  cis^  acquires 
once  all  its  possible  values  corresponding  to  real  values  of 
$,  when  6  passes  continuously  from  o  to  27r  and,  k  being 
any  integer,  repeats  the  same  cycle  of  values  through  every 
interval  from  2kTr  to  2  (/^ -f-  i)7^>  so  that  for  all  integral 

values  of /^, 

cis  {9  4-  2/(:7r)  =  cis  9. 

In  consequence  of  this  property,  cis^  is  said  to  h^periodiCy 
having  t\\Q  period  2  7r. 

The  exponential  B^  has  similarly  a  period.  Solving  for 
7C  and  V  the  two  linear  eqations, 

?/=  21  cos  /3  4-  ^  sin  /3, 

v'^=vcosiS  —  ?^sin/3,  (Art.  73.) 


we  easily  find 
whence 


7c^  cos  /3  —  v^  sin  /3 
z/  sin  ^  -}-  ^'  cos  /3 ; 


It  -J-  iv  =  (?<!'-f~  i"^^^  cis  /3  ; 

or  this  result  may  be  inferred  by  direct  inspection  of  Fig.  36. 
Hence  the  equation  ^"+'^  =  ^"'cis  27'/;;/,  of  Art.  73,  may 
be  written  ^ 

VI 

In  this  formula  let  71^=0  and  z''=  2kv1.1T,  k  being  any 
integer.     Then,  since  cis  2>^7r=  i, 

and  therefore 

Hence  -5^  has  the  period  27 kit.  In  particular  c'"  has  the 
period  2irr, 


92  THE    ALGEBRA    OF    COMPLEX    QUANTITIES. 

It  is  obvious  that  the  only  series  of  values  of  v^  that 
will  render  q\?>v' lm=^\  is  2k7mr,  where  k  is  an  integer 
(Art.  63).  Hence  the  only  value  of  w,  that  will  render 
Bw+w,^j^-a  ig  ic^=:z2kiKTr,  The  function  B"^  therefore 
has  only  one  period  and  is  said  to  be  singly  periodic. 

82.  Many-Valuedness  of  Logarithms.  As  a  con- 
sequence of  the  periodicity  of  B'^  =  z^  the  logarithm  (for 
all  integral  values  of  k)  has  the  form 

log^  Z=^W  ^  2ikKTZ, 

that  is,  log^-s,  for  a  given  value  of  z,  has  an  indefinitely 
great  number  of  values,  differing  from  each  other,  in  suc- 
cessive pairs,  by  2ZK7r.  The  logarithm  is  therefore  said  to 
be  many-valued ;  specifically  its  many-valuedness  is  infinite. 
We  shall  discover  this  property  in  other  functions.  In  the 
natural  agonic  system, 

111  ^7=  W  +    ^.ikTT. 

83.  Direct  and  Inverse  Processes.     In  the  present 

section  and  in  Sec.  VII  we  have  had  occasion  to  speak  of 

logarithms   and   exponentials    as  inverse  to  one  another. 

More  explicitly,  the  exponential  is  called  the  direct  function, 

the  logarithm  its  inverse.      We  express  the  operation  of 

inversion  in   general  terms   by   letting  f  (or  some  other 

letter)  stand  for  any  one  of  the  direct  functional  symbols 

used,  such  as  B,  or  ^,  thus  expressing  5-  as  a  direct  function 

of  w  in  the  form 

z=f(w), 

and  then  writing 

for  the  purpose  of  expressing  the  fact  that  w  is  the  corre- 
sponding inverse  function  of  z.  Thus  when  B''^  takes  the 
place  ofy(tC'),  log^  B  takes  the  place  of  /"'  (5-). 


THE  ALGEBRA  OF  COMPLEX  QUANTITIES.       93 

In  Other  terms,  inversion  is  described  as  that  process 
which  annuls  the  effect  of  the  direct  process.  If2=/(w), 
the  effect  of  the  operation  /  is  annulled  by  the  operation 
/-%  thus: 

In  accordance  with  this  definition  the  following  processes 
are  inverse  to  each  other : 

(i).      Addition  and  subtraction  : 

(x  -|-  a)  —  a  =  X,     (x  —  a)  -\~  a  =  x. 
(ii).      Multiplication  and  division: 

(x  >(  a)  I  a  =^  X,     (.r  /  a)  ><  a  -~  x, 
(iii).      Exponentiation  and  logarithmic  operation  : 

(iv).      Involution  and  evolution  : 

{^x°yia  =  x,   (x^'^Y=x. 

In  the  use  of  the  notation  of  inversion  here  described,  its 
application  to  symbolic  operation,  in  the  form//""'  (^w)  =  ic, 
must  be  carefully  distinguished  from  its  application  to 
products  and  quotients  by  which  from  ah=^c  is  derived 
a=^b-^c. 

Whenever  the  direct  function  is  periodic  its  inverse  is 
obviously  many-valued;  for,  if  /  be  a  period  oi  f(w'), 
so  that 

then 

y*""'  (2')  =  ze;  -f-  7lp, 

for  all  integral  values  of  ?^.  (Cf  Art.  82.) 


94 


THE   ALGEBRA    OF    COMPLEX    QUANTITIES. 


84.     Agenda.     Reduction  of  Exponential    and    Loga- 
rithmic Forms.      Prove  the  following  : 

^^u  /2-in(a=+3=)-z.tan-^  b/a  ^ig  [iv\n  {a' -^  b')  +  u  tan"^^/^] . 
(^il^yc-Tiv  ^  ^«]n3-z/-/2cls  (z;ln^  -i-  ^  -mi). 

log„i+/n  {'^-\-ij')  =  \  m  In  (.r"  +_>/=)  —  71  tan-^j'/x 

logm+in iy  =  VI  \\\y  —  \mr  -\-i(^n  \\\y  -^  \  vni)? 
lo&j  ( —  x)  =  zlnx  —  TT. 
log_i  ( —  x)  =  —  Inx  —  ZTT. 


(2) 
(3) 
(4) 
(5) 

(6) 
(7) 
(8) 
(9) 


logz^' 


■^^      l0gz(-2)  = 


(10).      Given  a -\~  zb  as  the  base  of  a  system  of  loga- 
rithms, find  the  modulus  and  reduce  it  to  the  form  7^.  -f  zv. 

(11).      Express  ^+^'^log  (.r -|- ?»  in  the  form  u -{- zv  in 
terms  of  a,  b,  x  andj^'. 

(12).      If  7/=  7^  cos /3 -j- z;  sin /3,   z'' =  z;  cos /3  —  7^  sin -3, 
2v=^zc-\-  iv  and  k  =-.  m  (cos  /3  +  i sin  /3),  prove : 


7;2 


^/ 


-  (e"'/'^  —  ^— «'/'^)  =  sinh  —  cos  —  +  7  cosh  —  sin  — 

2  ^  ■'  VI  VI     ■  VI  VI 

(13).      Vxovci  f{w')=^aw'' ^  2bw  ^  c=^z  deduce 


/-^  {2•)  =  {-b±,^    b^-ac-\-  az)la. 


THE   ALGEBRA    OF   COMPLEX    QUANTITIES.  95 

XIII.     What  Constitutes  an  Algebra? 

85.     The    Cycle    of  Operations   Complete.     It  is 

obvious  from  the  manner  in  which  in  the  preceding  pages 
the  algebraic  processes  as  appUed  to  complex  quantities 
have  been  defined,  that  out  of  their  operation  in  any  possible 
combination  no  new  forms  of  quantity  can  arise.  But  in 
particular  the  examples  of  Art.  67  illustrate  this  fact  in  an 
expHcit  way  in  the  case  of  geometric  addition  and  multipli- 
cation ;  and  tlie  exponential  equation 

„   ,  .        ,        p  ,     •   o      .   ^  cos  ,3  —  7c  sin  /3 

J^ic  +  tv  — —  1)11  COS  ii+vswiji  •  Qjg  ! L_  , 

m 

(Art.  73.) 
and  the  logarithmic  equation 

^og^(x  -{-  (v)  -=  Khi  (x  -j-  ly)  (Art.  75.) 

=  Kln(TA^-+7-^^) 
=  ^  In  (a-  +  j.=)  +  k  In  cis  (tan-^^J  , 

or  log^  (.r  +  /»  =  2  In  (^^  +  r)  +  ^''<  tan-^l » 

exhibit  with  equal  clearness  that  only  complex  quantities 
can  result  from  the  application  of  exponential  and  loga- 
rithmic operations.  Hence  the  processes  of  addition  and 
subtraction,  multiplication  and  division,  involution  and 
evolution,  exponentiation  and  the  taking  of  logarithms 
complete  the  cycle  of  operations  necessary  to  algebra. 
Other  operations  indeed  will  be  introduced  and  applied  to 
complex  quantities,  such  as  those  defined  in  Art.  90,  but 
though  expressed  in  abbreviated  form  as  single  operations, 
they  are  combinations  of  those  already  described. 


96  THE    ALGEBRA    OF    COMPLEX    QUANTITIES. 

86.  Definition  of  an  Algebra.  When  a  series  of 
elements  operating  upon  each  other  in  accordance  with 
fixed  laws  produce  only  other  elements  belonging  to  the 
same  series,  they  are  said  to  constitute  a  group.  Thus  all 
positive  integers,  subject  only  to  the  processes  of  addition 
and  multiplication,  produce  only  positive  integers,  and 
hence  form  a  group.     Such  a  group  is  an  algebra. 

The  effect  of  introducing  into  the  arithmetic  of  positive 
integers  the  further  processes  of  subtraction  and  division  is 
to  break  the  integrity  of  the  old  group  and  form  a  new  one 
whose  elements  include,  not  only  positive  integers,  but  all 
rational  numbers,  both  positive  and  negative,  integral  and 
fractional.  A  final  step  through  evolution,  or  the  extract- 
ing of  roots,  bringing  with  it  the  logarithmic  operation, 
leads  to  imaginary  and  complex  numbers  —  that  is,  numbers 
composed  of  both  a  real  and  an  imaginary  part. 

If  now,  as  is  legitimate,  we  regard  all  reals  and  imag- 
inariesas  special  forms  of  complex  quantities — reals  having 
zero  imaginary  parts,  imaginaries  having  zero  real  parts  — 
then,  as  pointed  out  in  the  preceding  article,  the  algebraic 
processes  of  addition,  subtraction,  multiplication,  division, 
evolution,  involution,  exponentiation,  and  the  taking  of 
logarithms  (logarithmication),  applied  to  complex  quantities 
in  any  of  their  several  forms,  produce  only  other  complex 
quantities.     And  hence : 

The  aggregate  of  all  complex  quantities  —  includijig  all 
reals  and  imagi7iaries^  both  7-ational  and  irrational —  ope7^at- 
ing  upoyi  each  other  in  all  possible  ways  by  the  rules  of 
algebra^  form  a  closed  group.  Such  a  groiip  is  again  an 
algebra. 

If,  in  an  algebra  the  elements  that  constitute  the  sub- 
jects of  its  operations  form  a  closed  group  when  subjected 
to  a  complete  cycle  of  such   operations,  such   an  algebra 


THE  ALGEBRA  OF  COMPLEX  QUANTITIES.      97 

may  be  said  to  be  logically  complete.  It  is  incomplete  if 
the  cycle  of  its  operations  be  not  a  closed  one.  An  algebra, 
for  example,  that  admits  evolution  and  the  logarithmic  pro- 
cess, but  precludes  the  imaginary  and  the  complex  quantity 
is  logically  only  the  fraction  of  an  algebra.  The  inability 
of  the  earlier  algebraists  to  recognize  this  fact  made  it  also 
impossible  for  them  to  carry  out  the  algebraic  processes  of 
evolution  and  the  taking  of  logarithms  to  any  except  real 
and  positive  numbers. 

With  this  fundamental  characteristic  of  the  algebra  of 
complex  quantities  the  reader  will  find  it  interesting  to 
compare  the  defining  principles  of  Peirce's  linear  associa- 
tive algebras,  outlined  in  his  memoir  on  that  subject,*  and 
Cayley's  observations  on  multiple  algebra  and  on  the  defi- 
nitions of  algebraic  operations,  contained  in  his  British 
Association  address  at  Southport  in  1883,  and  subsequently 
amplified  in  a  paper  published  in  the  Quarterly  Jouryial 
of  Mathematics  for  1 887.  *  * 


XIV.     Numerical  Measures. 

87.  Scale  of  Equal  Parts.  Every  magnitude,  real, 
ivtiaginaiy  or  complex,  as  represented  by  a  straight  li7ie,  can 
be  measicred  by  means  of  an  arbitrary  scale  of  equal  parts, 
called  units,  and  can  be  expressed  in  terms  of  the  assumed 
2cnit  by  a  rational  7mmber,  real,  imaginary  or  co^nplex,  with 
an  error  that  is  less  than  any  assignable  mwtber.  The 
method  may  be  exemplified  as  follows : 


*  Benjamin  Peirce :  Linear  Associative  Algebra  (Washington,  1870),  or  Ameri- 
can Journal  of  Mathematics^  Vol.  IV  (1881),  p.  97. 

**Cayley:  Presidential  Address  in  Report  of  the  British  Association  for 
the  Advancement  of  Science,  for  the  year  18S3,  and  on  Multiple  Algebra  in  the 
Quarterly  Jo7irnal of  Mathenialics  {iS'&-]),  Vol.  XXII,  p.  270. 


98      THE  ALGEBRA  OF  COMPLEX  QUANTITIES. 

88.  For  Real  Magnitudes.  Suppose  the  scale  of 
equal  parts  to  be  constructed,  of  which  the  arbitrary  unit 
shall  be 

OA^AB  =  BC= =  MN=j\ 

and  suppose  the  outer  extremity,  P,  of  the  given  magnitude 
OP  to  fall  between  the  vi^^^  and  (?;?  -{-  i)^^  points  of  division 

B  A'  O  A  B  C M  P       N 

—  2  — I  123  VI  m  +  i 

/ '^-  37- 

of  the  scale.  Then  7?i  is  the  integral  number  of  full  unit- 
lengths  contained  in  OP,  and  OM,  or  w  Xj\  differs  from 
OP  by  a  magnitude  that  is  at  least  less  than  yJ/A^  that  is, 
less  than  i  xy,  so  that 

OP=jXm-\-jX«i), 

where  (<  i)  means  'something  less  than  i.' 

Divide  the  segment  MJV  into  r  equal  parts  and  suppose 
P  to  fall  between  the  /i^^  and  (/i  -\-  i)^^  points  of  division. 
Then  k  is  the  integral  number  of  r^^  parts  of  the  unity  by 
which  OP  exceeds  OAf  and  the  excess  of  OP  over 
OM-^j  X  C-^^/^)  is  less  than  the  r^^^  part  ofy  ;  that  is 

op=yx(«  +  p)+yx(<y.)- 

Divide  the  r^^  part  of  MN  upon  which  P  falls  into  r 
equal  parts,  and  suppose  P  to  fall  between  the  kS^  and 
(^-j-  i)^^  points  of  division.  Then,  by  the  same  reasoning 
as  in  the  preceding  case, 


THE   ALGEBRA    OF    COMPLEX    QUANTITIES.  99 

If,  in  the  continuation  of  this  process,  P  eventually  falls 
upon  one  of  the  points  of  division  last  inserted,  OP  is 
exactly  measured  by  a  rational  number  of  the  form 


-'  +  -  +  -+      +^ 


But  it  may  be  that  the  process  will  never  end.  The 
successive  rational  numbers  will  then  be  more  and  more 
nearly  the  measures  of  OP,  but  never  exactly.  If  the  pro- 
cess be  continued  to  the  (^  +  i)*^'^  term,  it  is  obvious  that 
the  error  committed  in  assuming  the  rational  number  thus 
obtained  as  the  measure  of  OP  will  be  less  than  i/r^; 
that  is 

OP=.j  X  («  +  \+^  +  i  +  .  .  .   +  A)  +y  X  (<  i)  . 

and  by  increasing  q  sufficiently  this  error  ilr^  may  be  made 
less  than  any  number  that  can  be  assigned. 

If  the  scale  be  decimal,  r  is  lo  and  this  numerical 
measure  of  OP  assumes  the  form 

OP=j  X  (w  .  hJd  .  .  .  /)  +7  X  «  o.ooo  .  .  .  i), 

a  decimal  number,  m  being  the  integral  part  of  the  rational 
term,  p  and  i  occupying  the  q^^^  decimal  place. 

If  the  magnitude  be  negative,  the  number  which  is  its 
approximate  measure  will  be  affected  with  the  negative 
sign,  but  will  differ  in  no  other  respect  from  the  number 
that  measures  a  positive  magnitude  of  equal  extent. 

The  common  measure  of  the  two  magnitudes  OP  and 
y,  here  sought,  may  be  described  as  their  greatest  common 
measure  of  the  form  i/r-.  When  the  process  continues  ad 
infinitum,  they  have  no  exact  common  measure. 


lOO 


THE    ALGEBRA   OF    COMPLEX    QUANTITIES. 


89.     For  Complex  Magnitudes.     If  the  magnitude 
be  complex  it  is  of  the  form 

OA^iAP=x-^iy 

in  which  x  andjK  are  real.     By  the  process  just  described 

these  two  real  constituents 
may  be  separately  measured 
by  rational  numbers  in  terms 
of  an  arbitrary  unit,  with 
errors  that,  in  each  case,  are 
less  than  an  arbitrarily  small 
number.      Thus  let   vi  and  n 

be  the  rational  numbers  that  measure  x  andjj',  with  errors 

respectively  less  than  e  and  77;  then 


Fig.  3S. 


x^iy 


0  +  ^'«^)l 


'■x"-^-!(<7^)+'(<7ft7)( 


=yx(?«4-2>o+y 


■-J  X  (w  +  tn)  +y  X\<V^'  4-  r  '  cisc^J: 


where  cos  <^  =  cye^  -f  7]-,  sin  <^  ==  77  /  j/e^  +  r}%  and  conse- 
quently tan  <f>  =  r]  e,  or  <^  =  tan"^  (  rj  /e). 

Since  c  and  1;  are  separately  less  than  an  arbitrarily  small 
real  number,  so  is  -\/e-  -f-  '>?''  *  cis  <^  less  than  an  arbitrarily 
small  complex  number.  Hence  the  rational  complex 
number  m  +  2?i  is  the  measure  of  x  -}-  zy  within  any  re- 
quired range  of  approximation. 


CHAPTER    IV. 

CYCLOMETRY. 


XV.     Cyclic  Functions. 

go.  Definitions.^  The  general  cyclic  functions,  of 
which  the  circular  and  hyperbolic  ratios  of  Chapter  II  are 
special  forms,  are  defined  by  the  following  identities,  in 
which  w  and  w^  stand  for  complex  quantities  and 

K^7n  *  cis  /3 ^  modulus. 
(«).     sin^  w^'^ {e'"l'^  —  <?-'^/'^), 

(^).      tan^z£/^sin^ze'/cos^z£^, 

(^).     sec^  z£;  ^  I  / cos/^  ze/, 
(y").    csc^  a^  ^  I /sin^  ze/. 

These  are  read:  sine  of  o^,  cosine  of  ze^,  etc.,  with  respect  to 
modulus  K.  They  may  be  appropriately  called  viodulo- 
cyclic^  or  modo-cydic functions. 

When  K  =  /,  (^a)  and  (^)  assume  the  special  forms 

sin,- w  =  \(^ e''"  —  ^-^■^) ,    cos,- iu=r-\  {e''^  +  e-''^), 


*  Cf.  American  Journal  of  Mathematics,  Vol.  XIV  (1S92),  p.  192,  where  tenta- 
tive definitions  of  these  functions,  slightly  different  from  the  above,  were  e:iven. 


I02 


CYCLOMETRV. 


which  we  now  adopt  as  the  definitions  of  the  circular  sine 
and  cosine,  and  write 

sin  w ^ sin/  w^i /esc w, 
coszt:^^cos/ee^^  i/secze^, 
tan  w ^  tan/  w^i  j cot  ee'. 

When  K=  I,  («)  and  (<^)  become 

sin,  w=^\  ((?"'  —  ^~^),    coSj  w  =  \  i^e'"  -f  ^~"')' 

which  we  adopt  as  the  definitions  of  the  hyperbohc  sine 
and  cosine,  and  write 

sinh  ze^^sin^  ze^^  i/cschze-, 
cosh  Z2^^  coSi  w^i  /sech  zt', 
tanh  w  ^  tan^  w^ij  coth  w. 

By  comparing  these  definitions,  for  the  arguments  zo'  and 
iw  respectively,  remembering  that  i^=  —  i  and  i  z=  —  /, 
we  find  easily  the  following  important  relations  : 


(70 


sin  iw  =  i  sinh  w,  sinh  zze^  =  /  sin  w, 

cos  z'ze'  =  cosh  w,  cosh  /ze'  =  cos  u\ 

tan  z'zt;  =  /  tanh  u\  tanh  zze'  =  i  tan  ze', 

cot  iw  =  —  i  coth  z£^  coth  iw  =  —  i  cot  w, 

sec  ?*ze'  =  sech  w,  sech  /ze^  =  sec  w, 

CSC  2'ze'  =  —  i  csch  ze',  csch  /ze'  =  —  i  esc  ze'. 


91,  Formulae.  From  the  foregoing  definitions  of  the 
modocyclic  functions  are  deduced,  by  the  processes  in- 
dicated, the  following  formulae : 

From  (a)  and  (d)  by  addition  and  subtraction: 
( I ).      cos^  w  -r  K—^  sin^  w  =  e^'  '''^, 


(2)- 


coSi^Zc'  —  K— ^sm.- 


^^z 


CYCLOMETRY.  IO3 

The  product  of  (i)  and  (2)  gives 

(3).      cos^  w  —  K~-  sin^  7t7  =  I. 

By  dividing   (3)   successively    by  its  first  and  second 
terms  : 

(4).      I  —  K~-  tan^  zv  =  sec^  ic. 
(5).       COt^  IV  —  K~-  =  CSC"^  zv. 

From  («)  and  (^)  by  carrying  out  the  indicated  multi- 
plications, additions  and  subtractions  : 

(6).      sin^  (zc  ih  ^z'') 

=  sln^  zv  •  cos^  7£/'zt  coS;^  zv  '  sin^  zi>\ 

(7).      cos^(za  zh'W^) 

>■      =  cos^  Z£^ '  cos^  7£/  it  K~^  sin^  zv  '  sin^  zc^. 

From  (6)  and  (7)  by  making  zv^  =  zv\ 
(8).  sin^  2z^'=  2  sin^ze/ •  cos^ze', 
(9).      cos^  2zv^=  cos^  z£/  +  *^~~  sin^K  7^. 

From  (3)  and  (9)  by  addition  and  subtraction: 
(10).      cos^  2te^  —  I  =  2/<~=sin^z^^, 
(11).      cos^  2ZV  -|-  I  =  2  cos^  zv. 

From  (6)  and  (7)  by  division : 

tan^7£/  ±:  \.-AXi^zv^ 


(12).      tan^  (ze' di  ze'')  = 


f      \  4.    /       ,       /\       doXf^zv^K  =tan^7^/ 

(13).      QO\.,^{w  :^zv  ) —       ^  ^ 


I  ±L  cot,^  w  '  tan,^  zju 


I04  CYCLOMETRV. 

From  (12)  and  (13)  by  making  w^=^w\ 

r     \       ^  2tan^7/y 

(14).      tan^2Z£;  =  — ■ — _,/"    ^ — > 

(15).      cot;^  2r£/  =  ^  (cot^zc  + '^~*tan^zt/). 

From  the  two  forms  of  (6)  by  addition  and  subtraction: 

(16).     sin^  (7*7 -f  ze'') -|- sin^  (z£/ —  ze;') 

=  2  si 

(17).     sm.f^{iv -\- w'') — sin^  (?£;  —  w^) 
From  the  two  forms  of  (7)  by  addition  and  subtraction  : 

(18).        COS^  (t(7 -f  ^'')  +  COS^  (Z£/ —  Z£'') 

=  2  COS^  7l'  '  COS;(  7C>^, 

(19).      cos^  (r^  +  ^<^')  ~  cos^  (z^;  —  zt'') 

=  2K~''sin^  7i' '  sin^  za\ 

In  (16),   (17),  (18),   (19)  write  2  for  z<7  +  ze/',  2^  for 
w  —  zv^;  they  then  take  the  respective  forms : 

( 20).     sin^  2  -{-  sin^  2^ 

=  2  sin^  K^  +  ^'0  '  cos^  i  (-^  -  ^'). 

(21).      sin^^  —  sin^^''' 

=  2cos^4  (-3'-}-^')  -sin^-K^-  — ^'), 

(22).        COS^ -S' -f  COS^  2-' 

=  2  COS^  -2  (^  +  ^0  •  COS;,  K-^  —  ^0> 

(23).        COS;,  2-  —  COS;,  2-' 

=  2K~=  siU;,^  (2  -\-  2^)  '  siU;,^  (2  —  2^). 


CYCLOMETRY. 


105 


From  (12)    and    (13)    by  transposition  of  terms  and 
reduction: 

(24).      tan^  7C'  ±  tan^  7^' 

=  sin^  (za  zh  w^)  '  sec^  za  '  sec^  za^\ 

(25).      cot^  Z£' dz  cot^  a'' 

=  sin^  (w^±:  w)  '  csc^  w  '  csc^  w\ 

From  (i)  and  (2)  by  involution: 

(26).      (cos^  ze/'dr /^'"'sin^ze/')^^ 

=  cos^  Tvza^  ±  K~'  sin^  ww^, 

a  generalized  form  of  Demoivre's  theorem.     (Cf  Art.  74.) 
All  of  the  foregoing  formulae  pass  into  the  corresponding 

circular  and  hyperbolic  special   forms,   by   the   respective 

substitutions  k  =  z  and  k=  i. 

Making  the  proper  substitutions  from   (g-),    (/z),    (z), 

(7)'    {^)y    (0  i^  (6)   and    (7),    after   assigning   to  k  the 

values  /  and  i  successively,  we  obtain  : 

(27).     sin(zv  ±tw^) 

=:s'mw  '  cosh  w^±:  i  cos  w  '  sinh  za% 

(28).      cos(76' zb  ze£^') 

=  cos  7u  '  cosh  ze^'nF  z'sin  za  '  sinh  za^, 

(29).      sinh  (e£/ zb /?£/') 

=  sinh  zv  '  cos  zv^  ±z  i  cosh  zv  '  sin  zv^, 

( 30) .      cosh  ( 7C^  ziz  zzf'' ) 

=  cosh  TV  '  cos  w'  ziz  2*  sinh  zu  '  sin  ze/'. 

By    definitions  (a)  and   (/5)  and  the  definitions   of  cir- 
cular and  hyperboHc  sines  and  cosines  (Art.  90) : 

(31).     sin^e£/  =  K sinh z£J!k  =  zk  sin  za/'tK, 

(32).     cos^  za  =  cosh  zcj/k  =  cos  zc/zk. 


I06  CYCLOMETRY. 

92.     Agenda.     Prove  the  following  formulae 
(i).     sm^^7c>=: 


(2).      cos^^u>=  ±z^ — 


(3). 

(4). 


sin^  7u  4-  sin;^  7^' tan^  ^  (7v  -f-  ?£/') 

sin^  7£'  —  sin^  ?<:''       tan^  ^  (w  —  w' ) 

COS;^  w  +  cos^  w'  K=  cot^  J  (?£/  -4-  7<."''') 


(5).      — ''- j ^ — j  =  t2in^^(7£j  dzTjf'y 


(6).        ^^ "^ =  K-COt^i  (7C'  -^  7C'). 

COS,.  7<:'  —  COS,.  7a 


(7).       Sin^27£/ 


—  K  -  tan^  Z£/ 
(8 ).     sin^^  37£'  =  3  sin^  7C'  •  cos^  7v  -f  k-=  sin]^  7£'. 

(9).      cos^  37<.v  =  3  K--  cos^  7CI  '  s\n\  7a  -f-  cos]^  7£'. 

/      \     4.                    3  tan^  7£/  +  K~=  tan!.  7a 
(10).   tan,.-^  7£' =  ^^ '^ '^ 


.      N  secI-  7a 

(11).   sec^  2w 


r      \    ^        1  sin,^  7a 

(12).   tan^-|7('  = 


COS^  7£'  -j-  I 

(13).  By  making  k  =  /  and  /c=i  successively  in  the 
formulae  of  Art.  91  deduce  the  corresponding  formulae  of 
the  circular  and  hyperbolic  functions. 


CYCLOMETRY. 


107 


93.     Periodicity    of   Modocyclic    Functions.      In 

Art.  81  it  was  shown  that  all  exponentials  are  periodic 
having  the  period  2/K7r.  The  forms  e"^"^,  e^"""  and  t'^ 
have  therefore  the  periods  2ZK7r,  27r,  and  2zV  respectively; 
hence  also  the  modocyclic,  circular  and  hyperbolic  func- 
tions, whose  definitions  render  them  explicitly  in  terms  of 
gwiK^  ^w  and  e"  (Art.  90)  have  the  same  respective  periods, 
so  that,  \i  f  stand  for  any  one  of  the  symbols  sin^,  cos^, 
tan^,  etc.,  g  for  sin,  cos,  etc.,  and  h  for  sinh,  cosh,  etc., 
the  law  is,  n  being  any  integer :      For  modocyclic  functions 

f{u<  -\-  2niK-K)  z=f(w)^ 
for  circular  functions 

g(iw  -\-  2inr)=g{7v), 
for  hyperbolic  functions 

h  (Z£/  -f  21llTr)  =  h  (?£/). 


94.     Agenda.     Functions  ofSubmultiples  of  the  Periods. 

(i).  The  Period  iK-rr.  Show  that  tan^a/  and  cot^7£/ 
have  also  the  shorter  period  iK-rr. 

(2).  Range  of  Vahies.  Compute  the  following  table 
of  the  range  of  values  of  the  modocyclic  functions: 


w  = 

i-^--  to  t-Kr: 
2 

7K-  to  3'"'^^ 

2 

•^"^"   to    2/K7Z 
2 

sinK  ^(^ 

+  zKO  to   +iK 

+  7K    to    ±7/C0 

±/kO  to  — /K 

—  7k:  to  TtKO 

COSk  zh' 

+  1  to  ±0 

±0  to  —I 

~i  to  TO 

+  0  to  +1 

tan/c  w 

TikO  to  +//C00 

±7 /coo    to   +tKO 

+  //C0   to   +7KC0 

i/K  GO    to    +Z/CO 

cot  AC  7XJ 

+  i<x/K  to  ±tO/K 

-f/O/k:  to  ±ix/K 

t?oo/K  to  TzO/'/C 

T70//C  to  izoo/c 

secKza 

+  1  to  ±co 

irCO    to   —I 

—  I    to    +00 

+  00   to    +1 

CSCk  ?^ 

i:l<x/K  to   —i/K 

—  i/K  to  +:'oo/k 

+  /00//C  to  +///<: 

+  !'/<   to    +7  00/K 

I08  CVCLOMETRY. 

(3).  Tabulate,  as  above,  the  range  of  values  of  the 
circular  and  hyperbolic  functions. 

Prove  the  following  formulae : 

(4).     sin^  {^zu  d=  /ktt)  =  —  sin^  w, 

(5).       COS^^W  ±  IKTr)^  — cos^z^, 

^^^         .      //ktt  \ 

(6).      sin^  I ±:  wj=  IK  cos^  7U, 

.    ,                   liKir  \  ... 

(7).     cos^f zL  ?£^*  =  dz  z/f  sin^j  ee/, 

(8).      tan^  (  " -  zL  7iv)  =  ±  k^  cot^  w, 
(9).     cot^  \^  ±  '"')  =  ±  K-tan^Z£/, 

,       .         .        /  3ZK7r\ 

(10).   sm^lze^zh- j -- -h  zkcos^  ze/> 

,      .  /  3  z'ktt  \       . , 

(11).   cos^l^^^ ±  x^l= -h  z/KSin^  Z£/, 

(12).     Write  similar  formulae  for  all  modocyclic  functions 

3 /ktt/ 2  ±  w,  and  2 /ktt  zb  a^. 

95.  The  Inverse  Functions.  The  modocyclic, 
circular  and  hyperbolic  functions,  being  defined  in  terms 
of  exponentials,  are  direct;  and  corresponding  to  them, 
through  inversion  (Art.  83)  we  have  the  inverse  modo- 
cyclic, circular,  and  hyperbolic  functions. 


CYCLOMETRV.  lOQ 

Applying  the  operation  of  inversion  to  the  three 
equations 

2=:^  f  {w  -\-  27iiKTr')  =y(z£/), 

2  =g  {W  -f-  27^7^)       =g  (Z£/), 

z=-  h  {w  -f-  271177')   =  h  {w), 
of  Art.  93  we  have  the  corresponding  inverse  forms : 

y~'  (2')  =  ?£/  -|-  2;^ZK7^, 

g-^  (2-)  =^W  ^  271-7?, 
/2~'  (^z)  =  W  -{-  27lZ7r, 

in  which  n  is  any  integer  whatever.  Thus,  Hke  logarithms, 
the  various  forms  of  inverse  cyclic  functions  are  many- 
valued  to  an  infinite  extent. 

We  may  however  define  the  interval  over  which  the 
values  of  w  shall  range  in  such  a  way  as  to  make  it  one- 
valued  within  the  interval  considered.  Care  must  be  taken 
to  do  this  in  writing  the  formulae  of  inverse  functions. 

96.  Agenda.  Formulae  of  Inverse  Functions.  By 
inverting  the  corresponding  direct  formulae  prove  the  follow- 
ing, and  assign  in  each  case  the  interval  for  which  the 
formula  is  true. 


(i).      sin^'^f  =  cos^' y  I -f- *<  ''■^^• 
(2).      sin7  X  =  tan-'     .      .     _^    -^  • 

"|/  I    -f-  K     ^  X^ 

(3).      sin7  a- ±  sin-> 

=  sin";:'  {x  ^^l  -\-  K~^y-  ±y  \^ i  -\-  k~^ x^). 

(4).      s'm-'x±:sm-'j' 

=  cos-'  (i/(i  +  K-^'^0(i+K-=x)  zh  K-\xy). 


no  CYCLOMETRV 

(5).      cos-^r  zb  COS-' J 


cos^^  {.vy  ±  V(x'  —  I)  (y  —  i)), 


_  _       -^  —J' 

(6).      tan^'  X  ±  tan^>  =  tan^^  --_^^ 


(7).      cot^^^dtcot^'j/  =  cot^      ^, 


I  dz  AC  ""^  ,-ry 


(8).      2  sin^';f  =  sin";^' (2:r|/i  -j-  k~-x~), 
(9).      2sin~'jr=  cos'J^'  (i  -f  2/<~-ji."). 

(10).    2  cos^'.r  =  cos"^' (2jr- —  i). 

2  i* 
(11).    2  tan";:'  X  =  tan"'  — —-^r^ — -, ' 


(12).    2  tan^'  X  =  cos^'  ^  _  ^_/^^,  - 

(13).    2  tan-^  X  =  sin-'  ^  _  ^,3  y. ' 

(14).  Deduce  the  corresponding  formulae  for  inverse 
circular  and  hyperbolic  functions  by  assigning  to  k  the 
values  4"  ^'  ^i^d  —  I  successively. 

97.  Logarithmic  Forms  of  Inverse  Cyclic 
Functions.  The  inverse  cyclic  functions  are  obviously 
logarithmic.  We  may  obtain  them  as  such  in  the  following 
manner.  Let  21,  v,  x,  y  be  four  quantities,  in  general  com- 
plex, such  that 

u  =  log^  z>,    X  =  cos^  2i,  y  =  sin^  21. 
Then 

^u/K  __  ^,  __  QQ^^  2c  -L  K~'  sin^  26  =  X  ■-]-  K~^jy, 


CYCLOMETRY.  in 


x=  it  y  I  -h  K~j\    K    j=  ±y  X  —  I. 

(Art.  91,  3.) 


.  • .      sin^'j'  =  AC  In  (j'/k  ±  Vy/K^  +  i), 
and 

cos";^'  X  =  K  In  (x  dz  i/.r"  —  i  ). 

The  logarithmic  equivalent  of  tan~'  2  may  be  deduced  from 
the  definition  tan^w^K  {e^/>^  —  ^-"^/'^)  /  {f'/^  -f-  ^-«'/'<)  ==^, 
by  solving  this  equation  as  a  quadratic  In  ^'^'/'^.     Thus, 

tan^  w  =  ~^,/^:^~  =  2, 


whence 


K    2 


w  =  tm\J  2-  =  -  hi  -' 


Since  cot^w=  ijx,  the  corresponding  formula  for  cot^^'^  is 


K         K?  -|-  I 

cot^  2  :=  -  In — 

'^  2         KZ  —  I 


The  forms  for  sec;^'  x  and  csc^'j/  are  also  obtained  from 
those  for  cos';^':^  and  sin~'_y  by  changing  x  and^  into  ijx 
and  ily  respectively.     They  are 


sec^'^  =  K  In  (i/.r  ±:  ^'  ijx^  —  i}- 
csc~'j/  --^  K  In  (i/kj/  ±:  \/  iJK^y'  +  i)- 


112  CYCLOMETRY. 

The  Student  can  easily  verify  these  results  by  making  the 
transformations  independently. 

The  hyperbolic  forms,  obtained  from  the  foregoing  by 
putting  K  =  I  are  important,  and  are  of  frequent  application 
in  the  integral  calculus.  They  are  given  in  Art.  56.  As 
there  explained,  for  real  values  of  21  the  positive  sign  before 
the  radicals  must  then  be  chosen.  The  forms  for  the 
inverse  circular  functions,  got  by  putting  k  =  7,  are  less 
frequently  useful. 

98.     Agenda.     Prove  the  following,  n  being  an  integer: 
( I ).     sin-'  i  K  (^  -  2-')  =  cos-'  ^  (.3-  +  ^-) 

=  log^  2  -\-  2711 KTT. 
(2).       tan-'K(2-=—  l)/(<^^-[-   l)  =  \og^2-^2?l2K7r. 

(4).     cos^'  (-^"  4"  i)^)  ^  ^'^  (cos  /3  cosh"'  p  —  sin  /3  cos"'  o-) 
-|-  im  (sin  /3  cosh"'  p  -\-  cos  /3  cos"'  o-), 

where  

p=  zt  i/(^  +  lA=—  ^0.    «■=  dz  \/\s  —  ]/j'  —  x=), 
s  =  1  (^x^ -^ y -\-  i) ,    and    Krr=;;/cis/3. 

(5).     sin"'  (x  -\-  ry)  =  vi  (cos  /3  cosh"'  P—  sin  /3  sin"'  2) 
-\-  im  (sin  /3  cosh"'  P+  cos  /3  sin"'  2), 

where,  as  before,  k  =  m  cis  /3,  and 

P=iii|/'(5H-i/6^^-^0,    ^=±.x/(^S-VS-'-X^), 

.Y  =  (—  ^  sin  /3  4- jj/  cos  /3)  /  w,     Y=  (^  cos  /3  +  j/  sin  /3)  / ?«. 
(6).      Reduce  tan"'  {x  +  ?»  to  the  form  it  +  iv. 


CHAPTER    V. 

GRAPHICAL  TRANSFORMATIONS. 


XVI.     Orthomorphosis  Upon  the  Sphere. 

gg.  Affix,  Correspondence,  Morphosis.  Every 
complex  quantity,  defined  geometrically  by  a  vector  drawn 
from  the  origin  with  proper  length  and  direction,  de- 
termines uniquely  a  point  in  its  plane,  namely  the  extremity 
of  the  vector ;  and  conversely,  to  every  point  in  the  plane 
corresponds  one  and  only  one  complex  quantity.  It  is 
convenient  therefore  to  assign,  as  the  geometrical  repre- 
sentative, or  affix,  "^  of  a  given  complex  quantity,  a  point  in 
a  plane,  to  note  its  different  states  by  a  series  of  affixes, 
and  to  represent  a  continuous  change  in  it  by  a  line,  its 
path,  in  general  not  straight. 

This  relation  between  point  and  complex  quantity  is 
described  as  a  one-to-one  correspondejice,  and  the  spreading 
out  upon  the  plane  of  the  points  or  paths  of  a  varying 
complex  quantity  is  its  morphosis  in  the  plane,  or  its  planar 
morphosis. 

100.  Stereographic  Projection.  By  means  of  a 
stereographic  projection  we  may  establish  a  one-to-one 
correspondence  between  the  plane  and  a  sphere,  so  that  a 
point  upon  the  plane  determines  uniquely  a  point  on  the 
sphere,  and  vice  versa;  and  the  spreading  out  of  all   the 


C.  Jordan  :  Cottrs  cT Analyse  de  T ^cole  Polytechnique ,  Vol.  I,  p.  io6.  Art.  ii6. 


114  GRAPHICAL    TRANSFORMATIONS. 

points  or  lines  on  the  sphere  that  correspond  to  the  affixes 
or  paths  of  a  complex  quantity  in  the  plane,  will  be  its 
morphosis,  more  specifically  its  orthomorphosis^^^  upon  the 
sphere.  We  accomplish  the  transformation  in  the  following 
manner. 

We  place  the  sphere  with  its  center  at  the  origin  of 
complex  quantities  in  the  plane.  Regarding  the  plane  as 
fixed  in  a  horizontal  position,  all  projecting  lines  are  drawn 
from  the  upper  extremity  of  the  vertical  diameter  as  a 
center  of  projection  (Fig.  39  of  Art.  loi).  Then  if  from 
this  center  of  projection  a  straight  line  be  drawn  to  any 
point  in  the  plane,  it  will  cut  the  spherical  surface  in  one 
other  point.  The  two  points  thus  uniquely  determined, 
one  in  the  plane,  the  other  on  the  sphere,  are  said  to 
correspond  to  each  other,  or  to  be  correspoyiding  points. 
Thus  a  complex  quantity  may  have  an  affix  in  the  plane 
and  a  corresponding  affix  on  the  sphere. 

Points  in  the  plane  inside  the  great  circle  in  which  it  cuts 
the  sphere  correspond  to  points  on  the  lower  hemisphere; 
points  in  the  plane  outside  this  circle  have  their  corre- 
spondents on  the  upper  hemisphere. 

loi.  Transformation  Formulae. ^^^  In  order  to  con- 
nect algebraically  the  two  forms  of  the  complex  quantity, 
in  the  plane  and  on  the  sphere,  assume  as  the  origin  of 
co-ordinates  in  both  systems  the  center  of  the  sphere,  let 
i  and  y)  be  the  horizontal,  t  the  vertical  co-ordinates  of 
points  on  the  sphere,  x  and  y  the  co-ordinates  of  corre- 


*An  orthomorphic  transformation  of  the  plane  into  the  spherical  surface. 
This  knid  of  transtormation  is  called  orthomorphosis  by  Cayley  :  Jotirnal  fur 
die  teine  utid  atigewandte  Mathematik,  Bd.  107  (1891),  p.  262,  and  Quarterly 
Journal  0/  Mathematics,  Vol.  XXV  (1S91),  p.  203.  See  also  the  first  footnote  to 
Art.  108. 

**  Cf .  Klehi :    Vorlesungen  iiber  das  Ikosaeder,  p.  32. 


GRAPHICAL   TRANSFORMATIONS. 


115 


spending  points  in  the  plane.  It  will  involve  no  loss  of 
generality  to  assume  the  radius  of  the  sphere  to  be  i.  Its 
equation  then  is 


Fig'  39- 

and  OC  being  the  vertical  axis,  P  and  O  corresponding 
points  on  the  plane  and  sphere  respectively,  we  have,  by 
similar  triangles, 


whence 


and  therefore 


X  _0P  _}>__      I 
^  V 

I  _  ^    ^     I  —  ^ 


zy 


Il6  GRAPHICAL   TRANSFORMATIONS. 

From    these    and    the    equation   of  the  sphere  we   readily 
obtain 


.T'+y+I 


2 


and  thence  the  values  of  ^,  i  and  rj  in  terms  of  x  and  j/, 
namely : 

.  2X 


v 

102.  The  Polar  Transformation.  If  it  be  desired 
to  present  the  formulae  of  transformation  in  terms  of  tensor 
and  amplitude,  we  may  write 

x  =  r  cos  6,     y=^r  sin  Q, 
I  =  cos  </)  •  cos  B,     rj^  cos  cf)  '  s'mO,     1  =  sin  (f>, 


and  accordingly 

sm  <l> 


cos<^        _    . 

:r  +  z  v  =  r  CIS  c^  = ^ — r     cis  c', 

•^  I  —  sm  o  ' 


cos<^ 


I  —  sin  ^ 

Thus  the  expression  cos<^  (i  —  sin  <^)  "cis^,  in  which  <^ 
and  0  are  independent  of  each  other,  suffices  to  represent 
all  possible  complex  quantities. 

By  easy  substitutions,   I,  rj,  t,  are  found,   in  terms  of  r 
and  0,  to  be 

.       2rcos^  2rsin^         r^  —  i 

^  ""   r^  +  i"  '    '^  ""  r^+  I  '    ^  ""  r^-f-  i  ' 


GRAPHICAL    TRANSFORMATIONS.  1 17 

103.  Agenda.  Properties  of  the  Stereographic  Pro- 
jection. 

(i).  Prove  analytically  that  a  circle,  or  a  straight  line 
in  the  plane,  corresponds  to  a  circle  on  the  sphere. 

(2).  Prove  geometrically  that  any  two  lines  in  the  plane 
cross  each  other  at  the  same  angle  as  the  corresponding 
lines  in  the  sphere.      (Cf.  Art.  107.) 

(3).  Show  that  to  the  centre  of  projection  correspond 
all  points  at  infinity  in  the  plane,  and  that  it  is  therefore 
consistent  to  say :  there  is  in  the  complex  plane  but  one 
point  at  infinity. 

(4).  Show  that  meridians  on  the  sphere  through  the 
centre  of  projection,  and  parallel  horizontal  circles  on  the 
sphere  correspond  respectively  to  straight  lines  through  the 
origin  and  concentric  circles  in  the  plane. 

XVII.     Planar  Orthomorphosis. 

104.  W-plane  and  Z-plane.  In  the  graphical  rep- 
resentation of  an  equation  connecting  two  complex  varying 
quantities  w  and  z,  it  conduces  to  clearness  of  delineation 
and  exposition  to  separate  the  figures  representing  the 
variations  of  lu  and  z,  and  to  speak  of  the  a/-plane  and  the 
^--plane  as  though  they  were  distinct  from  one  another. 
This  language  and  procedure  help  us  to  see  more  clearly 
that  the  plane  with  the  ze^-markings  upon  it  has  a  distinctive 
character  and  presents  in  general  an  appearance  different 
from  that  which  it  has  when  its  markings  represent  the 
variations  of  the  functions  of  w,  and  to  distinguish  more 
easily  the  two  groups  of  markings  from  one  another. 

It  is  the  purpose  of  the  present  section  to  describe  the 
planar  orthomorphosis  of  some  of  the  functions  that  have 
been  defined  in  the  foregoing  pages,   that  is,  to  cause  the 


Il8  GRAPHICAL    TRANSFORMATIONS. 

point  Q,  the  affix  oiw,  to  traverse  the  ze'-plane  in  a  specified 
manner,  and  to  mark  out  the  paths  that  P,  the  affix  of  a 
function  of  w,  will  follow,  in  consequence  of  the  assumed 
variations  of  w. 

105.  The  Logarithmic  Spirals  of  B^  Non-inter- 
secting. The  function  ^"',  is  singly  periodic;  that  is, 
there  is  only  one  quantity,  the  period  siktt,  multiples  of 
which,  when  substituted  for  w,  will  render  B'^  =  \,  (Art. 
81).  If  now  Q^  and  Q,  the  affixes  oiw^  and  w,  move  in  the 
ze^-plane  upon  parallel  straight  lines,  the  variable  quantities 
Wq  and  w  may  be  assumed  to  have  the  relation 

w  =  w^  -f-  «, 

where  <«  is  a  constant  quantity  (fixed  in  length  and  direc- 
tion);* and  the  paths  of  >5^'o  and  B'""  will  either  not  inter- 
sect at  all,  or  will  coincide  throughout  their  whole  extent. 
For,  in  order  that  the  two  paths  may  have  a  point  in 
common  there  must  be  a  pair  of  values  w^,  zi'^-j-a,  for 
which 

and  a  must  be  a  multiple  of  2ZKTr  (Art.  81).  But  if  a  be 
a  multiple  of  p/ktt,  then  for  all  values  of  za,^ 

and  the  two  ze'-curves  have  all  their  points  common.  Hence, 
since  in  the  construction  of  Fig.  36,  Art.  68,  a  vector 
representing  21  kit  must  lie  in  the  direction  01^,  we  conclude : 
Tf  the  paths  of  w^  and  w  in  the  w-plane  be  parallel 
straight  liiies,  the  paths  of  B'''  and  B'"^  in  the  z-plane  will 


*This  is  merely  a  way  of  saying,  that  if  a  link,  or  rod,  while  remaining 
parallel  to  a  fixed  direction,  move  with  one  of  its  extremities  upon  a  fixed 
straight  line,  its  other  extremity  generates  a  second  straight  line  parallel  to  the 
first. 


GRAPHICAL   TRANSFORMATIONS. 


119 


be  coincident,  or  distinct  and  not  intersecting,  according  as 
the  intercept  viade  by  the  two  w-lincs  on  the  modnlar  7ior- 
vial  is  or  is  not  a  vmltiple  of  the  period  ^/ktt. 

106.  Orthomorphosis  of  B^.  The  fixed  elements 
in  the  z<7-plane  are  the  real  axis,  the  modular  line  and  the 
modular  normal, — in  Fig.  36,  the  lines  O/,  ET  and  OF; 
in  the  ^--plane  they  are  the  real  axis  and  the  unit  circle. 

By  the  operation  of  exponentiation,  indicated  by  ^^, 
a  straight  line  in  the  ze'-plane  is  transformed,  metamorphosed, 
into  a  logarithmic  spiral  (Art.  80).  Hence  if  the  variable 
elements  of  the  ze'-plane  be  assumed  to  be  straight  lines,  in 
the  ^'-plane  they  will  be  logarithmic  spirals. 

Assigning  as  the  path  of  w^  a  straight  line  OS^  passing 
through  the  origin  (Fig.  40),  write 

W=W^'\-  aiK, 

in  which  a  is  a  real  quantity.  The  path  of  w,  for  a  given 
value  of  a,  will  then  be  a  line  EC,  parallel  to  OS^,  to 
which  will  correspond  in  the  ^--plane,  a  logarithmic  spiral 
E'C^  (Fig  41).  In  particular  to  the  path  of  w^  corres- 
ponds the  spiral  z,^  that  passes  through  the  intersection  of 
the  real  axis  with  the  unit  circle. 


w-plane. 


Fig.  40. 


z-plane. 


Fig.  41. 


I20  GRAPHICAL    TRANSFORMATIONS. 

To  the  straight  lines  in  the  zu-plane,  obtained  by  giving 
different  values  to  a,  there  correspond  in  the  ^--plane,  so 
long  as  a  is  less  than  2-  and  not  less  than  o,  distinct  non- 
intersecting  logarithmic  spirals  (Art.  105).  And  since 
OC=c=  ma  (Art.  72),  when  a  varies  from  o  to  2-,  c 
varies  from  o  to  2  m  tz,  or  as  represented  in  Fig.  40,  from  o 
to  (9 C  and  when  ^C  moves  from  the  position  OS^  to  the 
position  £iSj ,  the  corresponding  logarithmic  spiral  makes 
a  complete  revolution  and  sweeps  over  the  entire  -S'-plane. 

To  every  point  in  the  ^--plane  corresponds  one  and  only 
one  point  in  the  band  between  the  parallels  OS^,  BiS,, 
whose  width  is  2W7rcos  (<^  — /3),  and  also  one  and  only 
one  point  within  every  band,  in  the  zc-plane,  having*  this 
width  and  parallel  to  OS^.  The  construction  therefore 
shows  graphically  how,  to  every  value  of  2,  there  cor- 
respond an  infinite  number  of  values  of  w,  namely  all  the 
values  w  -}-  2k2KTr,  wherein  k  is  any  integer.  The  suc- 
cessive affixes  of  ze^,  w  -\~  qikit^  w  —  iiktv,  w  -f-  4Z'<7r,  etc., 
are  obviously  situated  at  the  division-points  of  equidistant 
intervals,  each  equal  to  2imr,  along  a  straight  line  through 
the  affix  of  w  parallel  to  OF. 

Whenever  a  z£/-line  crosses  OF,  the  corresponding 
^■-spiral  crosses  the  unit  circle  (Art.  68).  Hence,  to  points 
in  the  z£^-plane  below  or  above  the  modular  normal  OF, 
correspond  respectively  points  in  the  ^--plane  within  or 
or  wfthout  the  unit  circle.  Thus  the  shaded  and  unshaded 
portions  of  Figs.  40,  41  correspond  respectively. 

The  points  where  any  spiral  in  the  ^'-plane  crosses  the 
real  axis  are  those  for  which  B'"^  is  real.  But  the  necessary 
and  sufficient  condition  for  the  vanishing  of  the  imaginary 
term  of  B'^"  is 

.    V  cos  /3  —  u  sin  /3  ,  „  ^ 

sm ^-— ^  =  0,  (Art.  73.) 


GRAPHICAL    TRANSFORMATIONS.  121 

for  which  purpose  it  is  necessary  and  sufficient  that 

V  cos  /3  —  7c  sin  /3  =  /t;;^7^, 
or 


It  cos 


( ^  +  /^)  +  ^^  sin^^  4-  /3j  ==  ^WTT, 


wherein  k  is  any  integer ;  and  the  locus  of  the  equation  last 
written  is  a  straight  line  parallel  to  0  7"  and  distant  from  the 
origin  a  multiple  of  mir.  Hence,  whenever  a  z£^-path  crosses 
such  a  line  the  corresponding  ^■-spiral  crosses  the  real  axis. 
(Cf.  Arts.  72,  80.) 

As  particular  constructions  we  have :  When  the  w-Ymes 
are  parallel  to  the  modular  line  O  T,  the  ^--spirals  degenerate 
into  straight  lines  passing  through  the  origin;  and  when 
the  ze^-lines  are  parallel  to  the  modular  normal  OF,  the 
^--spirals  become  circles  concentric  with  the  unit  circle. 
(Cf.  Art.  71.) 

107.  Isogonal  Relationship.  Any  two  spiral  paths 
of  B'^  cross  one  another  at  the  same  angle  as  the  correspond- 
ing straight  paths  of  w. 

Let  B^,  p^y  v^,  7C^  and  B,  p,  v\  u^  be  two  sets  of  cor- 
responding values  of  0,  p,  v^  and  it'  (Fig.  36).  It  was 
shown  in  Art.  72  that  mO  =  v^ ;  hence 

and  Ml:zlol  _  A  .  ^JJUlo . 

P-Po  ^'^     P—Po 

But  (Fig.  36) 

v'—  v^  =  (u'—  u.^)  tan  (<^  —  /3), 
whence,  smc& p  =  b"'   and  p,^=zb"o  (Art.  73), 

A(^-^o)  -P:  .  -'<=i^a  tan  (d.  _  ,3). 


122 


GRAPHICAL    TRANSFORMATIONS. 


and  in  Art.  52  it  was  shown  that 


limit    \ b'''  —  b"o\      p^  . 
limit    \p,  (^  -  ^  J 


/=/oi     P-P. 


=  tan(<^-/3). 


In  Fig.  42  let  7^^/*  represent  a  segment  of  the  spiral  of 


B'",    P^T  2.   tangent    line   at    P^ 
and  let 

P^,p=OPr_,,    OP, 

6^  =  arc- ratio  of  XOP,^, 


z-plane. 


0  =  arc-ratio  of  XOP, 


Then 


/  -P,=  HP  and  p^  {B  -  0^)  =  P^//; 


whence 


limit    \pj^ 
P=Po 
that  is, 


=1 


tan  (arc-ratio  of  KP^  7"), 


arc-ratio  of  KP-^  T=  <f>  —  /3. 

Thus  the  angle  between  the  radius  vector  p  and  the 
spiral  is  constant  and  equal  to  that  between  the  path  of  zc 
and  the  modular  line  (Art.  68).  It  follows  that  the  angle 
between  any  two  spirals  is  the  same  as  that  between  the  two 
corresponding  paths  of  w.     O.  E.  D. 

It  follows  also,  that  to  a  small  triangle  formed  by  any 
three  ze^-lines  in  the  ze^-plane  corresponds  a  small  curvilinear 
triangle  in  the  ^--plane,  whose  angles  are  respectively  equal 
to  those  of  the  former  and  whose  sides  become  more  and 
more  nearly  proportional  to  those  of  its  correspondent  as 


GRAPHICAL    TRANSFORMATIONS.  I23 

the  two  triangles  become  smaller.  The  two  figures  there- 
fore approach  the  condition  of  similarity  to  one  another  as 
a  limit,  when  they  themselves  approach  the  vanishing  point. 
It  is  because  of  this  property  that  the  transformation  of 
of  w  into  B^  is  called  orthomorphic.^^  It  is  a  property 
that  can  be  proved  to  be  generally  true  of  functions  of  a 
complex  variable.  ^-'^ 

108.     Orthomorphosis  of  cos^(u -h  iv.)     In  Art.  8 1 
it  was  shown  that 

u  -\-  iv  =  (?/-}-  ry)  cis  /3, 

and  it  is  obvious  from  the  definitions  of  Art.  90  that 

.  w  .        w 

cos^  w  =  cosh  -   =  cosh  — -• — ^  • 
'^  K  ni  CIS  p  ' 

whence 

cos^  in  -\-  2V)=^  cosh  — — —  • 

Here  u^  and  v^  have  the  significations  attached  to  them  in 
Art.  68,  Fig.  36.     For  brevity  write 

?//  VI  =  r,    v^l  m  =  s. 

By  formula  (30)  of  Art.  91  we  have 

cosh  (r  -[-  ?V)  =  cosh  r  cos  s  ^  i  sinh  r  sin  s. 
Let 

ji:  ^  cosh  r  cos  i^,   j/ ^  sinh  r  sin  j-; 

then  X-       ,        v' 

-j—    .   —  —  J 

cosh^  r      sinh"  r        ' 

and  X'  y- 

cos  ■  s      sin"  s 


*"Orthomorphic.  .  .  .  Preserving  the  true  or  original  shape  of  the  infini- 
tesimal parts,  though  it  may  be  expanding  or  contracting  them  unequally." 
( Cenijay  Dictionary. ) 

**Byerly:  Integral  Calculus,  2d  ed.,  Art.  211,  p.  275. 


124  GRAPHICAL    TRANSFORMATIONS. 

• 

In  these  equations  x  and  y  are  the  Cartesian  co-ordinates 
of  the  varying  affixes  of  cos^^  (?^  -f-  iv'). 

As  equations  of  loci  they  make  the  following  correlations 
apparent :  When  r  remains  constant  and  j-  varies,  the  affix 
of  21  +  iv  describes  a  straight  line  parallel  to  the  modular 
normal  while  the  affix  of  cos^  (?^  +  z<^0  moves  upon  an 
ellipse;  when  s  remains  constant  and  r  varies,  the  affix  of 
u  -f-  iv  describes  a  straight  line  parallel  to  the  modular 
line  while  the  affix  of  cos^  (?/; -f  z^)  moves  upon  a 
hyperbola. 

109.  By  Confocal  Ellipses.  The  former  of  the  last 
two  equations  represents,  for  r=  constant,  an  ellipse 
whose  principal  semi-diameters  are  cosh  r  and  sinh  r.  Since 
coshV  — sinh-r=  i,  any  two  or  more  of  the  ellipses  ob- 
tained by  giving  different  values  to  r  are  confocal,^  having 
their  foci  on  the  real  axis  at  unit's  distance  on  either  side  of 
the  origin. 

If  r  be  regarded  as  a  parameter  and  be  allowed  to  vary 
from  o  to  +00,  the  ellipse  represented  by  the  equation 
.T7cosh-r-T-_>/"/sinh-r=  I  ,  starting  from  the  segment  of 
straight  line  joining  the  foci  as  its  initial  form,^-^  expands 
and  sweeps  over  the  entire  plane.  Since  the  equation  is 
the  same  whether  r  be  positive  or  negative,  the  variation  of 
r  from  o  to  —  00  gives  a  like  result.  It  is  only  necessary, 
therefore,  in  order  to  make  x  andjv  pass  once  through  all 
real  values,  to  retain  the  positive  values  of  r. 

In  any  one  of  the  ellipses  the  variations  oi  x  and  y  are 
governed  by  the  variations  of  cos  s  and  sin  j ;  real  values  of 
x  lie  between  —  cosh  r  and  -f-  cosh  r,  real  values  of  y 
between  —  sinh  r  and  +  sinh  r.      To  the  values 


*  Charles  Smith:     Conic  Sections,  Art.  221,  p.  244. 


GRAPHICAL    TRANSFORMATION'S. 


125 


O,      -TT,       TT,      -TT, 


2  7r, 


correspond  respectively 

X  =  cosh  ;•,        o,        —  cosh  ;',  o,  cosh ;-, 

y  =      o,        sinh  ?',  o,  —  sinh  ;-,        o. 

By  comparing  these  limits  of  variation  we  easily  dis- 
cover, r  remaining  constant  and  positive,  that  when  s  passes 
through  each  successive  quadrant  of  a  unit  circle,  the  point 
{x,  y)  describes,  in  the  same  order,  the  successive  quad- 
rants of  an  ellipse.  If  r  be  negative,  the  same  ellipse 
reappears,  but  its  periphery  is  described  in  the  reverse 
direction.  Thus  the  affix  of  cosh  (r  -f-  'is')  describes  the 
entire  ellipse  when  s  passes  from  o  to  27r;  or  obviously  also, 
when  .c  passes  from  any  value  s^  to  Sq~\-  27r. 

If  r  and  s  be  now  replaced  by  u^  j  7ii  and  v^ j  vi  and 
cosh(r+?V)  by  cos^{u  -\-  iv),  the  results  of  this  ortho- 
morphosis  may  be  collated  as  follows : 

When  u^  varies  from  o  to  -f  00  and  v^  from  some  fixed 
value  v^  to  v^-^  2imr,  the  affix  of  cos^iji  -\-  iv)  ranges 
over  the  entire  ^--plane;  that  is,  to  the  band  A  A  in  the 
a/-plane    (Fig.    43),    of  width   2imr,    extending   from  the 

TV-plane.  z-plaii(\ 


Pis.  43- 


Fig.  44- 


1^6  GRAPHICAL    TRANSFORMATIONS. 

modular  normal  OF  upwards  to  infinity  parallel  to  the 
modular  line,  corresponds  the  entire  s-plane  (Fig.  44). 

In  like  manner  to  the  band  BB,  below  the  modular 
normal,  and  also  to  every  other  rectangular  band  con- 
gruent to  AA  having  its  base  in  the  modular  normal, 
corresponds  the  entire  2'-plane. 

no.  By  Confocal  Hyperbolas.  By  an  analysis 
similar  to  that  of  Art.  100  we  derive  from  the  equation 

cos-i"       sin-^ 

a  series  of  confocal  hyperbolas,  having  cos  s  and  sin  s  as 
their  principal  semi-axes  and,  since  cos* «? -f  sin- ^  =  i , 
having  their  foci  at  the  points  -f  i  and  —  i,  and  forming 
therefore  with  the  ellipses  of  Art.  109  a  series  of  confocal 
conies.  -'^ 

Any  hyperbola  of  the  series  may  be  generated  by  giving 
to  s  the  four  values  2it  —  s,  s,  tt  -f  ^,  it  —  s  \x\.  succession 
and  in  each  case  allowing  r  to  vary  from  o  to  +  oc  ;  .  the 
values  2  7r  —  s  and  s  produce  one  branch  of  the  hyper- 
bola, IT  -^  s  and  IT  —  s  the  other.  If  then,  in  addition  to 
the  variations  of  ?',  s  pass  continuously  from  s^  to  s-^  -\-  ^tt, 
the  series  of  hyperbolas  will  pass  over  the  whole  of  the 
z-plane  and  the  affix  of  cosh  (r  -f  is)  will  range  over  the 
band  A  A,  Fig.  43.  This  co-ordination  of  the  two  figures 
is  thus  the  same  as  that  established  in  Art.  109. 

But  the  co-ordination  may  take  place  in  other  ways.  It 
is  possible,  for  example,   to  generate  one  entire  branch  of 


*  As  is  well  known,  these  confocal  conies  cut  one  another  orthogonally.  fSee 
Charles  Smith's  Conic  Sections.  Art.  224,  p.  246.)  In  other  words,  the  transforma- 
tion of  u-\-iv  into  cos,^  («+2z/)  is  orthomorphic ;  and  in  virtue  of  this  ortho- 
morphism,  when  the  path  oi  ti-\-iv  is  any  straight  line,  that  of  cos^  (z<+zz/)  is  a 
spiral  that  cuts  the  ellipses,  and  also  the  h>-perbolas,  at  a  constant  angle. 


GRAPHICAL    TRANSFORMATIONS.  I  27 

any  hyperbola  of  the  series  by  keeping  s  fixed  and  varying 
r  fi-om  —  CO  to  -]-  ^ )  -^  passing  from  cos  ^  to  -f-  00 ,  ^  from 
—  00  through  o  to  -f-  co.  The  other  branch  of  the  same 
hyperbola  is  then  obtained  by  changing  «?  into  tt  -\-  s  and 
causing  ?^  to  vary  again  from  —  cc  to  -\-  cc  .  If  then,  in 
addition  to  the  variation  of  r,  s  be  given  all  values  between 
s^  and  s,^  -{-  i-n-,  the  morphosis  is  completed.  To  the 
2'-plane  in  this  case  are  co-ordinated  two  parallel  bands,  a  a 
and  b  b  o{  Fig.  43,  having  a  width  equal  to  \in'K  and 
extending  to  infinity  in  both  directions,  and  separated  from 
one  another  by  a  third  band  of  like  dimensions. 

III.  Agenda.  Problems  in  Orthomorphosis.  De- 
scribe the  graphical  transformation  of  it  -\-  iv  into  each  of 
the  following  functions  : 

(i).      sin^  {2C  +  hi).  (2).      tan^  {ic  -f  iv). 

(3).      sec^(/^  +  ?V).  (4).      {it -{-ivy. 

(5).  Prove  that  the  graphical  transformation  of  tt^  iv 
into  cos^  {it  -f  zV),  or  into  any  of  the  other  cyclic  functions 
is  an  orthomorphosis. 

(6).  Perform  ypon  ^("+'^)/'<^,  sm^{u-\-  iv),  cos^(?^-f- /z') 
and  tan^  {u  -|-  iv),  the  transformation  of  Art.  loi  and  trace, 
upon  the  surface  of  the  sphere,  the  path  of  each  of  these 
functions  when  the  affix  of  2t  -j-  iv  moves  upon  a  straight 
line  in  the  plane. 


CHAPTER    VI. 

PROPERTIES  OF  POLYNOMIALS. 


XVIII.     Roots  of  Complex  Quantities. 


112.     Definition  of  an  n^^  Root.     The  n^'^^  root  of 


a 


given  quantity  is  defined  to  be  such  another  quantity  as, 
when  muhiplied  by  itself  ?^  —  i  times  (used  n  times  as  a 
factor),  will  produce  the  given  quantity.  An  7i^^  root  of  zc 
is  denoted  by  w^^'^ 

Throughout  the  discussion  concerning  roots,  whether  of 
quantities  or  equations,  n  is  supposed  to  be  an  integer,  and 
unless  statement  be  made  to  the  contrary,  a  positi\'e  integer. 

113.  Evaluation  of  n*^  Roots.  Every  complex 
qua^itity  has  n  7i^^^  roofs  of  the  form 

,      .     2kTr-^6 

11  ' 

in  which  ?'^/"  is  a  tensor,  {ikir  ^-  6)  j  11  an  amplitude,  and 
k  has  one  of  the  values  o,  i,  2,    .  .  .  ,   7^  —  i. 

Let  the  complex  quantity,  whose  roots  are  to  be  in- 
vestigated, be  denoted  by  r  cis  0.  Since  r  is  a  real  positive 
magnitude,  r^^'"  has  one  real  positiv^e  value  (Art.  23).  How 
to  find  this  value  we  do  not  here  enquire. 

By  Demoivre's  theorem  (Art.  74),  for  all  integral  values 
ofX^ 

\  rv«  cis  ^-^ I    =  r  cis  (  2X^7r  +  ^)  =  r  cis  0, 


PROPERTIES    OF    POLYNOMIALS.  1 29 

and  the  complex  quantities 

,    .  e      ,    .  217  ^-  e  ,    .  2(11  —  1)17-^6 

^i/n  CIS  -  ,    r'/"  CIS  -  -  --    > ,    r'/''  CIS  -^ J—1-. 

n  n  n 

are  all  different.  Hence  each  of  them  is  a  distinct  7^^^'  root 
of  rcis^,  and  there  are  n  of  them.  Thus  rcis^  has  n  n^^ 
roots,  which  was  to  be  proved. 

No  additional  values  are  derived  from  r^/«  cis  {2kir  -f  (f)ln 
by  giving  to  k  any  values  other  than  those  contained  in  the 
series  o,  i,  2,  .  .  .  .  7^—  i,  and  r^/"  has  only  one  real 
positive  value.      Hence 

A  complex  qiiantity  has  only  ii  /^^'^  7'oots. 

114.  Agenda.  Examples  in  the  Determination  of  z?*^ 
PvOOts.      Prove  the  followmg:-'^ 

(2).     The  cube  roots  of  —  i  are  — i  and  \  ±i\  y^2>' 

(3).      The  fourth  roots  of  —  i  are  iti  — 7-  and  ±  — ^— • 

^^^  1/2  1/2 

(4).      The  cube  roots  of  i  -\-  i  are 

-1+^    (l/3_±„0±i(j/3ziO, 

2^/3         '  24/3 

and  -  ( 1/ 3_-_ i)  ±i_(j/ 3_±0  . 

24/3 

(5).     The  sixth  roots  of  4-  i  are 

±  I,    ^  ±:  ii  1/3    and    -\±  i\  1/3. 


*Cf.  ChrystaL   Algebta.  Vol.  1,  pp  242,  248. 


130  PROPERTIES    OF    POLYNOMIALS. 

(6).      Find  the  fifth   roots  of  -f  i   and  the  sixth  roots 
of—  I. 

(7).      If  w  be  one  of  the  complex  cube  roots  of  -f  i, 
then 

( I  -l-  (o-y  =  <ji,   I  -|-  o)  -f  o>-  =  o, 

and 

(i  —  oj  -f-  u)-y=  (^1  -\-  w  —  o)-)>  =  —  8. 

Show  that  I  -r  <^"is  one  of  the  twelfth  roots  of  -f  i. 

(8).      The  twentieth  roots  of  +  i    are    the    successive 
powers,  from  the  first  to  the  twentieth  inclusive,  of 

ih' (10 +  2^/5) +  ^'(1/5 -I)  J. 

(9).      Representing  the  complex  number  of  example  (8) 
by  0-,  show  that 

0-'  —  o-"  4-  o-^  —  o-=  +  I  =  o,    o-"  +  I  =  o, 

and  that  or  is  therefore  a  tenth  root  of  —  i.     Show  that  the 
even  powers  of  cr  are  the  tenth  roots  of  -f-  i- 


XIX.     Polynomials  and  Equations. 

115.     Definition  of  Polynomial.     An   algebraic  ex- 
pression of  the  form 

a^^a,z-f-  a,2^  -\-  .  .  .  .  -j-  a„  2", 

in  which  «q,  «,,  .  .  .  an  are  any  quantities  not  involving  2, 
and  in  which  the  exponents  of  2  are  all  integers,  is  called  a 
rational,  integral  polynomial  iii  z.  In  what  follows  it  will 
be  sufficient  to  speak  of  it  more  briefly  as  a  polynomial,  the 
qualifying  adjectives  being  understood.  The  highest  ex- 
ponent oi  z  contained  in  it  is  its  degree. 


PROPERTIES  OF  POLYNOMIALS.  131 

116.  Roots  of  Equations.  The  investigaaon  of 
Art.  113  solves  the  problem  of  finding  what  are  called  the 
roots  of  the  equation 

^71    --    r^^ 

in  which  ze^  is  a  known  complex  quantity,  and  shows  that  such 
an  equation,  which  would  commonly  present  itself  in  the 
binominal  form 

az''  4-  <^  =  o,  \bla^=  —  w'X 

has  exactly  n  roots. 

If  additional  terms  containing  powers  oi  z  lower  than  the 
71^^  be  introduced  into  this  equation,  the  problem  of  its 
solution  becomes  at  once  difficult,  or  impossible.  In  fact, 
the  so-called  algebraic  solution  of  a  general  algebraic  equa- 
tion of  a  degree  higher  than  the  fourth,  that  is,  a  solution 
involving  only  radicals  and  having  a  finite  number  of  terms, 
is  known  to  be  impossible.^ 

A  discussion  of  the  methods  that  may  be  employed  in 
solving  equations  is  beyond  the  intended  scope  of  the 
present  work,  but  the  so-called  fundamental  theorem  of 
algebra  (Art.  120),  accompanied  by  those  propositions  that 
are  prerequisite  to  its  demonstration,  find  a  fitting  place 
here. 

117.  The  Remainder  Theorem.  If  a  polynojnial 
of  the  71^^  degree  in  z  be  divided  by  z  —  y,  the  remainder, 
after  n  successive  divisimis,  is  the  result  of  substituting  a  for 
z  in  the  polynomial. 

Let /"(£-)  denote  the  polynomial. 


*  Proved  to  be  so  by  Abel;  Journal  fur  die  reine  und  angewandte  Mathematik 
(1826),  Bd.  I,  pp.  65-84,  and  CEuvres  covipietes  de  N.  H.  Abel,  nouv.  ed.,  Vol.  I,  pp. 
66-94. 


132  PROPERTIES    OF    POLYNOMIALS. 

and  let  the  division  by  2  —  y  be  performed.  It  is  obvious 
that  the  remainder  after  the  first  division  will  be  of  a  degree 
lower  by  i  than  the  dividend,  that  each  succeeding  re- 
mainder will  be  of  a  degree  lower  by  i  than  its  predecessor, 
and  that  therefore  the  71^^  remainder  will  not  involve  z.  If 
the  final  quotient  be  denoted  by  O  and  the  final  remainder 
by  R,  then 

z  —  y        ^    '    3"  —  y 
in  which  R  does  not  involve  z\  whence,  by  multiplying  by 

fiz)  =  Q(z-y)  +  R. 

This  equation  has  the  properties  of  an  identity,''^  and 
in  it  z  may  therefore  have  any  value  whatever.  Accord- 
ingly, let  y  be  substituted  for  z,  and  let  the  result  of  this 
substitution  in  the  polynomial  be  denoted  byy(y);  then 

/(y)=-^(7-y)  +  ^, 

in  which  R  remains  unchanged  from  its  former  value, 
and   Q,   being  now   a  polynomial    in  y,    is  finite.      Hence 

and  ^-/(y).  Q.  E.  D. 

If  the  remainder  obtained  in  dividing /" (5')  hy  z  —  y 
vanish,  then  /(y)  ^o  and  y  is  said  to  be  a  root  of  the 
equation/C^-)  =  o.      Hence: 


*This  may  be  shown  by  actually  evolving  it  in  specific  instances.     Thus,  if 
the  process  here  described  be  applied  to  the  quadratic  a^-^-a^z-Va,^'^,  the  result  is 

and  the  principle  of  this  procedure  is  obviously  general,  and  independent  of  the 
degree  of  the  polynomial.  It  should  be  observed  that  the  identity  does  not  depend 
upon  the  process  of  division;  we  might,  in  fact,  produce  it  by  the  processes  of 
addition,  subtraction  and  rearrangement  of  terms.  The  di\  ision  process  is  used 
as  a  convenience,  not  by  necessity. 


PROPERTIES    OF    POLYNOMIALS.  133 

(i).  If  fi^^^  be  exactly  divisible  by  z  —  y,  y  is  a  I'oot  of 
the  equation  f  {^z')  =  o. 

Conversely,  the  remainder  will  vanish  ify(y)^o. 
Hence : 

(ii).  jfy  be  a  root  of  the  equation  f(^z)  =^  o,  thenf(^z) 
is  exactly  divisible  by  z  —  y. 

118.  Argand's  Theorem.  If  for  a  given  vahie  of 
z  the  polynomial  of  the  n^^^  degree, 

a^-\-a,z-^  a.z''  -\-  .  .  .  .  -j-  a„  z", 

have  a  value  w  ^  different  from  zero,  its  coefficients  a^,  a^, 
a^,  ...  an  being  given  quantities,  real,  iinagiiiary,  or  com- 
plex, there  exists  a  second  value  of  z,  of  the  form  x  -f-  y', 
for  which  the  polynomial  has  another  value  w  such  that 

tsr  w  <  tsr  w^. 

The  following  demonstration  of  this  theorem  is  a  modifi- 
cation of  Argand's  original  proof.* 

Let  Zq  be  the  given  value  of  5-  and  let  w^  be  the  resulting 
value  of  the  polynomial,  different  from  zero,  so  that 

1^^  =  a^-\-  a^z  -^  a^z"  -{-  .  .  .  .  +  ^,,  z". 

Add  to  Zq  an  arbitrary  complex  increment  z,  whose  tensor 
and  amplitude  are  disposable  at  pleasure,  and  let  w  be  the 
resulting  value  of  the  polynomial,  so  that 

7£/  =  ^,  +  ^,(0^  +  s)+«,(0^  +  0)=+   .    .    .    +^»G-o  +  ^)"- 

If  the  several  powers  of  the  binominal  z^  -j-  z,  in  this 
equation  be  expanded  by  actual  multiplication,  or  by  the 


*Argand:  Essai  sur  nne  manicre  de  reprcsenter  les  quantites  imaginaires 
dans  les  constructions  geometriques  (Paris,  1806;,  Art.  31.  The  demonstration 
was  reproduced  by  Cauchy,  in  \.he  Journal  de  V Ecole  Royale  Pidytechnique  (1S20), 
Vol.  XI,  pp.  411-416,  in  his  Analyse  algebrique  (1821),  ch.  X,  and  again  in  his 
Exercises  d^  analyse  et  de  physique  mathnnatique.  Vol.  IV,  pp.  167-170. 


134  PROPERTIES    OF    POLYNOMIALS. 

binominal  theorem,  and  its  terms  be  then  arranged  accord- 
ing to  the  ascending  powers  of  z,  it  may  be  written  in  the 
form 

7^'  =  ^o  +  ^i^o  +  ^3^!)  +  ^3^o-r  .  •  .  ^a^zl 

in  which  b^,  <^,.,  ,  .  .  bn_^  involve  z^  but  not  z.  By  hypothesis 
a„  is  not  zero,  but  any  or  all  of  the  coefficients  b^,  b^, 
.  .  .  b„_^  may  possibly  vanish.  Let  b„i  be  the  first  that  does 
not  vanish,  b,i  standing  for  the  same  thing  as  a,i,  so  that 

7U  =  7C'^-^  b„,Z"^  -f  {b„,+  ,  -}-  K  +  22^ +  b„Z"-"^-^)  Z"^+\ 

and  let  b„i  =  a  cis  a, 

bm-ri^bm+2Z-\-   .    .    .    +-^.,2"-'«-^=^cis/3, 

and 

s  =  r  cis  6, 

the  quantities  a,  a,  b,  /3,  r  and  B  being  real.      Then 

w  =  7i'^  -[-  ar"'  cis  (a  -f  ;;/  0)  -f  ^/-"'+^  cis  )  /3  -r  (^'^  —  i )  ^  J  • 

Since  the  length  of  any  side  of  a  closed  polygon  cannot 
be  greater  than  the  sum  of  the  lengths  of  all  the  other  sides, 
or  in  other  words,  since  the  tensor  of  a  sum  cannot  be 
greater  that  the  sum  of  the  tensors  of  the  several  terms 
(Art.  58), 

.  • .     b<  tsr/^„,+i  -I-  r  tsr  b„,^^  -f  .  .  .  -f  ^«-;;^-I  tsr  ^„. 

Hence,  by  diminishing  r  sufficiently  b  may  be  made  to 
differ  from  tsr^,„+i  by  an  arbitrarily  small  quantity,  and  a 
maximum  limit  to  the  variation  of  r  may  be  assigned  such 
that  b  shall  not  exceed  a  fixed  finite  value.  It  follows  that 
r  may  be  taken  so  small  that 

br<^a,    or    br'"^'' <ar"'. 


PROPERTIES    OF    POLYNOMIALS. 


135 


Let  F^,  P'  and  P  (Fig.  45)  be  the  respective  affixes 
of  w^,  w^-^  ar''^  cis  (a  -f  vid)  and  za,  and  let  0  and  r,  which 
are  at  our  disposal,  be  so  chosen  that 

a  -f-  mO  =  amp  w^-{-  it, 
and  ar'"  <  OP^  ; 

and    if  this    disposition    of  r    be    not   sufficient   to    make 
l^ytn+i  ^  ^yvt^  ig|-  ^  ]3g  s|-jii  further  diminished  until 

<5r  <  « 
Then,  since  cis  (amp  tu^)  ^=  vsr  w^  (Art.  60), 
w  =  zu^—  ar'"^  cis  (amp  0/3)  +  br^^^^  cis  J  /3  +  (;;2  +  i )  ^  j 
=  (tsr  Z£/^  —  ar'"')  vsr  w^  +  br"'"^^  cis  f  /3  +  {in  +  i )  ^  j , 
and  ^;.w^+i  <-  ^^m  ^  i-si-  ^^^ 

In  accordance  with  these  relative 
determinations  of  tensors  and  am- 
plitudes the  positions  of  P^,  P^ 
and  P  in  the  ze'-plane  (Fig.  45) 
are  as  follows : 

/*Q,  the  affix  of  iv^,  lies  upon  a 
circumference  whose  center  is  O  and 
radius  OPr.. 


Fig.  45- 


P\  the  affix  of  (tsrz^/^— d;r^«)  vsrz£/Q,   lies    upon    OP^ 
between  O  and  P. 

P,  the  affix  of  w,  lies  upon  a  circumference  whose  centre 
is  /"  and  whose  radius  is  less  than  P^  P^. 

This  latter  circumference  is  therefore  wholly  within  that 
upon  which  P^  lies,  and  P  is  nearer  to  O  than  is  Pj^.     But 


hence 


OP^^tsvw^   and    OP- 
tsrTc;  <  tsra/,^. 


tsrw 


O.  E.  D. 


136  PROPERTIES    OF    POLYNOMIALS. 

When  all  the   coefficients  of  z  in   the  expansion   of  w 

except  an  vanish    (^bn=^a,t),  the  point   F'    is    coincident 

with  P.      The  final  result  in  the  foregoing  demonstration 
remains. 

119.  Every    Algebraic    Equation    has    a    Root. 

There  exists  at  least  one  value  of  2,   real,   imaginary,   or 
complex,  for  which  the  polynoinial  of  the  n^^^  degree, 

a^-ra,z-^a._2^^  .  .  .  +^„-a", 

variishes,  the  coefficients  a  ^,  a^,  ...  aa  being  given  q2ia7itities, 
real,  imaginaiy  or  complex. 

This  theorem  is  commonly  stated  in  the  briefer  form: 
Every  algebraic  equation  has  a  root.  It  is  an  immediate  con- 
sequence of  Argand's  theorem.'^  For,  if  it  be  assumed 
that  there  is  no  value  of  z  for  which  the  polynomial 
vanishes,  and  if  w^  be  that  value  of  the  latter  whose  tensor 
is  the  least  possible,  this  hypothesis  is  at  once  contradicted 
by  Argand's  theorem  which  asserts  that  there  is  another 
value  of  z  and  a  corresponding  value  of  zu  such  that 

tsrzt'  <  tsrzc'Q, 

Hence,  tsr  w  must  ha^•e  zero  as  its  least  value,  and  for  such 
a  value  the  polynomial  vanishes. 

120.  The    Fundamental    Theorem    of  Algebra. 

A  polynomial  of  the  n^^  degree,  snch  as 

a^^  a,z  -^  a^z^-  -^  .  .  .  Ar  cin z'\ 

whose  coefficioits  a^,  a^,   .  .  .  an  are  given  real,  imagijiary, 
or  complex  quantities,   is  equal  to  the  product  of  n  linear 


*  The  two  propositions  were  not  segregated  by  Argand;  both  were  proved  by 
him  in  the  same  paragraph  {loc.  cit  Art.  31}. 


PROPERTIES    OF    POLYNOMIALS.  1 37 

factors  multiplied  by  the  coefficient  of  the  highest pozver  of  z 
171  the  poly noviialS^ 

Let  y"(2')  stand  for  the  polynomial.  By  Art.  119 
y(2')  =  o  has  at  least  one  root;  let  that  root  be  y^.  Then, 
by  Art.  117,  /"(^O  is  divisible  by  z  —  y^  and  may  be  ex- 
pressed in  the  form 

in  which/,  (2)  is  a  polynomial  of  the  degree  n  —  \.  The 
equation  /(2')  =  o  has  also  a  root;  let  this  root  be  y.. 
Then,  as  before, 

/.(^)  =  (^-y,)/,(^), 
in  which/  (2-)  is  a  polynomial  of  the  degree  n  —  2  ;  whence 

This  process  repeated  n  times  will  produce  a  quotient  of 
the  degree  71  —  n^=o,  that  is,  a  quantity  independent  of  z. 
Hence,  /(-s-)  may  be  expressed  as  the  product  of  71  linear 
factors  and  a  factor  independent  of  z;  and  because  the 
coefficients  of  the  highest  powers  of  z  on  the  two  sides  of 
our  equation  must  be  identical,  this  last  factor  is  «„.      Thus 

/(2')  =  an  (2-  —  yO  (2-  —  y,)  .  .  .  .  (.-  —  y,). 

O.  E.  D. 


*The  first  demonstration  of  this  celebrated  theorem  was  given  by  Gauss  in 
his  Doctor-Dissertation  which  bore  the  title  Dctnonstratio  nova  theoreniatis 
omneni  functio7iem  algebraicum  rationalem  integrant  unius  variabilis  inf adores 
reales primivel secuiidi gradns  resolvi posse,  and  was  published  at  Helmstiidt  in 
1799.  He  presented  a  second  proof  in  December,  1815,  and  a  third  in  January, 
1816,  to  the  Kuiiigliche  Gesellschaft  der  Wissenschaften  zu  Gottingen.  (Sec 
Gauss:  IVerke,  Bd.  III.)  F"or  further  notices  concerning  this  theorem  see  the 
following:  H.  Hankel:  Complexe  Zahlen,\ip.^-j-g'&\  Burnsideand  Panton:  Theory 
0/  Equations,  2d  ed.,  pp.  442-444;  Baltzer:  Element e  der  Mathematik,  6.  Aufl., 
Bd.  I,  p.  299. 


138  PROPERTIES    OF    POLYNOMIALS. 

As  a  corollary  of  this  theorem  we  have :  A71  equation  of 
the  tiS^  degree  has  n  and  only  n  ?'oots.  For,  the  condition 
necessary  and  sufficient  in  order  that  /(^)  may  vanish,  is 
that  one  of  its  linear  factors  shall  be  zero,  and  the  putting 
of  any  one  of  its  linear  factors  equal  to  zero  gi\es  one  and 
only  one  value  of -S".  Thusy(-S')  will  vanish  for  the  n  values 
Vu  72  >   •  •  •  >  y«  of  2-,  and  for  no  others. 

121.  Agenda.  Prove  the  following  theorems  con- 
cerning polynomials : 

( I ).  Every  polynomial  in  x  -f  iy  can  be  reduced  to  the 
ioxmX-YiY. 

(2).  If/(jir4-t;0  be  a  polynomial  in  x -^  iy  having 
all  its  coefficients  real,  and  if 

f{x-\-7y)  =  X-^iY, 
then 

f{^x-iy')=^X—iY. 

(3).  If  all  the  coefficients  of  the  polynomial  y(-3')  be 
real,  and  if 

f{a^ib)-=o, 
then 

f(^a  —  id)  =  0. 

(4).  In  an  algebraic  equation  having  real  coefficients 
imaginary  roots  occur  in  pairs. 


APPENDIX 


SOME    AMPLIFICATIONS. 

Art.  23,  page  40. 

The  notation  b^  does  not  here  presuppose  any  knowledge 
of  the  fact,  easily  proved  as  indicated  in  Art.  38,  that  by  has 
the  usual  arithmetical  meaning  when  b  andj/  are  numbers. 

Art.  24,  page  42. 

The  following  systematic  arrangement  of  the  steps  of 
the  proof  of  the  first  proposition  of  page  42  will  aid  the 
student  to  a  clearer  apprehension  of  it. 

First  System.  Second  System. 

base  =  b,  base  =  c, 

modulus  =  m,  modulus  =  km, 

y  =  \og„i  X,  ky  =  \ogkm  X, 

by  =  X,  c^y  =  X, 

x^=b  when_y  =1,  x=^c  when  j/  =  ijk, 
.-.     b^'k=,c   and   c^=^b. 

Art.  27,  pages  43-44. 

The  following  alternative  proof  of  the  law  of  indices, 
though  longer  than  that  given  on  pages  43-44,  has  the 
merit  of  greater  explicitness : 


140 


APPENDIX. 


Let  the  straight  line  SP,  while  remaining  transversal  to 
the  two  intersecting  straight  lines   OP,  OS,  move  parallel 
to  a  fixed  direction  with  a  speed 
proportional  to  its  perpendicular 
distance  from  O.      Since   OP  and 
OS  are  proportional  to  this  dis- 
tance,   P  and   6^  move  also  with 
speeds  respectively  proportional  to 
to  their  distances  from  O. 
Let   Q  move  along  a  straight  line  through   O  with  a 
constant  speed  /x,  and  let  the  following  sets  of  values  corre- 
spond respectively,  as  designated  by  the  accents : 

OP,    0P\     OP''=x,  x\  x'\ 
OS,     OS',     OS''  =  s,    s\    s'\ 

00   OQ'    0Q''=y,  y,  y\ 

Finally,  let  OS'  and  OJ  be  each  equal  to  the  linear  unit 

and  let 

speed  of  P  at  /=  A.. 

If  t  represent  the  time  it  takes  SP  to  pass  to  the  position 
S''P'' ,  then  designating  ratio  of  distance  traversed  to  time- 
interval  as  average  speed  (abbreviated  to  av.  sp.),  we  have 

av.  sp.  of  6*=  (  /'  —  s)  /  f, 
av.  sp.  ofP=lx''-x)/t, 
and 

(av.  sp.  of  S)  I  (av.  sp.  of  P)  =  (s''~  s)  /  (x''-  x). 

But  s''/s  =  x'\/x, 

that  is,  (/'-  s)/s=(x''-  x)/x, 

or  (s''—  s)  I  (x''—  x)^=sj X, 


av.  sp.  of  6' 
av.  sp.  of  P 


APPENDIX. 


141 


This  proportion  remains,  however  small  the  interval  be, 
and  the  limit  of  average  speed,  as  the  interval  approaches 
zero,  is  the  actual  speed  at  the  beginning  of  the  interval. 
It  follows  that 

speed  of  S s speed  of  S 

speed  of  P      x  A  .r 

whence 

speed  of  S=  \s, 

and  speed  of  S  at  unit's  distance  from  O  is  A..  But  by  the 
definition  of  an  exponential  (Art.  23),  since  the  positions 
0\  Q"  correspond  respectivelv  both  to  P\  P^'  and 
t^  S\   S'\ 

x'=  exp„,y,    x''=exp,ny', 

and,  since  0S^=  i, 

s''=exp,„(y'—y), 

where  m  =  fx  /  A.      But  again,  since  s^=  i, 

.  • .     exp„,y'  j  exp„,y'=  exp,,,  ( j''  — J'')- 


END   OF  THE  UNIPLANAR   ALGEBRA. 


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